## Abstract

We investigate the way to control multi-wave mixing (MWM) process in Rydberg atoms via the interaction between Rydberg blockade and light field dressing effect. Considering both of the primary and secondary blockades, we theoretically study the MWM process in both diatomic and quadratomic systems, in which the enhancement, suppression and avoided crossing can be affected by the atomic internuclear distance or external electric field intensity. In the diatomic system, we also can eliminate the primary blockade by the dressing effect. Such investigations have potential applications in quantum computing with Rydberg atom as the carrier of qubit.

© 2013 OSA

## 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 ^{85}Rb. The resonant wavelengths (frequencies) in this system are 795 nm (Ω_{1}≈Ω_{3} = 377.1 THz) from both of 5S_{1/2} (F = 3) and 5S_{1/2} (F = 2) to 5P_{1/2}, 762.1 nm (Ω_{2} = 393.3 THz) for 5P_{1/2} to 5D_{3/2}, and 474.2 nm (Ω_{4} = 632.18 THz) for 5P_{1/2} to 70S_{1/2}, respectively. If |4> is 69D_{3/2}, the resonant frequency will be Ω_{4} = 632.2 THz. The probe beam *E*_{1} (frequency *ω*_{1}, wave vector *k*_{1}, Rabi frequency *G*_{1} and frequency detuning Δ_{1}) connects the transition |0>−|1>; *E*_{2} (*ω*_{2}, *k*_{2}, *G*_{2} and Δ_{2}) and ${{E}^{\prime}}_{2}$ (*ω*_{2}, ${{k}^{\prime}}_{2}$, ${{G}^{\prime}}_{2}$ and Δ_{2}) drive |1>−|2>; *E*_{3} (*ω*_{3}, *k*_{3}, *G*_{3} and Δ_{3}) and ${{E}^{\prime}}_{3}$ (*ω*_{3}, ${{k}^{\prime}}_{3}$, ${{G}^{\prime}}_{3}$ and Δ_{3}) connect |1>−|3>; *E*_{4} (*ω*_{4}, *k*_{4}, *G*_{4} and Δ_{4}) drives |1>−|4>. The Rabi frequency *G _{i}* is defined as

*G*=

_{i}*μ*/

_{ij}E_{i}*ћ*, where

*μ*is the transition dipole moment between |

_{ij}*i*> and |

*j*>. The frequency detuning Δ

*is defined as Δ*

_{i}*= Ω*

_{i}*−*

_{i}*ω*.

_{i}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 *E*_{3} and ${{E}^{\prime}}_{3}$ are on, a FWM signal *E*_{F} dressed by strong beam *E*_{4} satisfying the phase-matching condition *k*_{F} = *k*_{1} + *k*_{2}−${{k}^{\prime}}_{2}$ can be generated in the |0>−|1>−|2> subsystem. While if all the beams except *E*_{2} and ${{E}^{\prime}}_{2}$ are on, a SWM signal *E*_{S} satisfying *k*_{S} = *k*_{1} + *k*_{3}−${{k}^{\prime}}_{3}$ + *k*_{4}−*k*_{4} 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, *ћ*Ω_{1}, 2*ћ*Ω_{1}, *ћ*Ω_{2}, *ћ*(Ω_{1} + Ω_{2}) and 2*ћ*Ω_{2}, 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 *ρ*_{10} (*ρ*_{1000} for diatomic system), which can be obtained by solving the density-matrix equations with perturbation method. First, when only *E*_{1}, *E*_{2} 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 ${G}_{F}=-i{G}_{1}{G}_{2}{({{G}^{\prime}}_{2})}^{*}\mathrm{exp}(i{k}_{F}\cdot r)$, *d*_{10} = Γ_{10} + *i*Δ_{1}, *d*_{20} = Γ_{20} + *i*(Δ_{1} + Δ_{2}) and Γ* _{ij}* being the transverse relaxation rate between |

*i*> and |

*j*>. Then, when all the beams except

*E*_{2}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 ${G}_{S}=i{G}_{1}{G}_{3}{({{G}^{\prime}}_{3})}^{*}{G}_{4}{({G}_{4})}^{*}\mathrm{exp}(i{k}_{S}\cdot r)$ and

*d*

_{30}= Γ

_{30}+

*i*(Δ

_{1}−Δ

_{3}). 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 *E*_{1}, *E*_{2}, ${{E}^{\prime}}_{2}$ and *E*_{4} 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 *d*_{40} = Γ_{40} + *i*(Δ_{1} + Δ_{4}). Because *E*_{4} does not participate in F1 process directly, we call it as *external-dressing* field. When *E*_{2} and ${{E}^{\prime}}_{2}$ are strong sufficiently, they will also bring dressing effect, that |10> will be first split into |*G*_{2 ±} 0> by *E*_{2} (${{E}^{\prime}}_{2}$), and then split into third energy levels |*G*_{2+}*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

*E*_{2}(${{E}^{\prime}}_{2}$) participate in F1 process directly, we call it as

*self-dressing field*. When ${E}_{4}$ is strong sufficiently, ${E}_{3}$ (${{E}^{\prime}}_{3}$) weak and ${E}_{2}$ kept off, the dressed perturbation chain for S1 process generating ${E}_{S}^{D}$ changes into (V) ${\rho}_{0000}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{{G}_{4}{}_{\pm}000}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{3000}^{(2)}\stackrel{{\omega}_{3}}{\to}$ ${\rho}_{{G}_{4}{}_{\pm}000}^{(3)}\stackrel{{\omega}_{4}}{\to}{\rho}_{4000}^{(4)}\stackrel{-{\omega}_{4}}{\to}{\rho}_{{G}_{4}{}_{\pm}000}^{(5)}$, by which we can obtain

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, Δ*E _{ns}*(

*R*)≈−

*C*

_{6}/

*R*

^{6}−

*C*

_{8}/

*R*

^{8}−

*C*

_{10}/

*R*

^{10}is for

*ns-ns*interaction, Δ

*E*(

_{np}*R*)≈−

*C*

_{5}/

*R*

^{5}−

*C*

_{6}/

*R*

^{6}−

*C*

_{8}/

*R*

^{8}for

*np-np*interaction, and Δ

*E*(

_{np}*R*)≈−

*C*

_{5}/

*R*

^{5}−

*C*

_{6}/

*R*

^{6}−

*C*

_{7}/

*R*

^{7}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 70*S*_{1/2}, and the corresponding energy shift of level |40> in the diatomic system can be written as Δ*E*_{70}* _{s}*(

*R*). With this energy level shift, the resonant frequency Ω

_{4}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

*D*

_{3/2}, to explore the property of Rydberg atoms more particularly. The energy level shift of $|40\u3009$ can be written as Δ

*E*

_{69}

*(*

_{d}*R*). Correspondingly the resonant frequency and frequency detuning are modified into ${{\Omega}^{\prime}}_{4}={\Omega}_{4}+\Delta {E}_{\text{69}d}(R)/\hslash $ and ${{\Delta}^{\prime}}_{4}={\Delta}_{4}+\Delta {E}_{69d}(R)/\hslash $. And the matrix density element for the dressed SWM process can be rewritten as

## 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 Δ_{1}<0, when Δ_{4} + Δ*E*_{70}* _{s}*(

*R*)/

*ħ*is around |

*G*

_{2+}0>, two secondarily split levels |

*G*

_{2+}

*G*

_{4 ±}0> could be created from |

*G*

_{2+}0> by the external-dressing effect as shown in Figs. 2(d1)-(d4). Symmetrically, in the region with Δ

_{1}>0, when Δ

_{4}+ Δ

*E*

_{70}

*(*

_{s}*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 Δ

_{4}+ Δ

*E*

_{70}

*(*

_{s}*R*)/

*ħ*resonant with the midpoint of |

*G*

_{2 ±}0>, the first and secondary dressing effects can together split |10> into |

*G*

_{2}

*G*

_{4 ±}0>, as shown in Fig. 2(d5).

First of all, the effect of Rydberg blockade can be revealed when Δ_{1} 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 Δ_{4} + Δ*E*_{70}* _{s}*(

*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 Δ

_{1}+ Δ

_{2}= 0) and |00>−|10>−|40> (appearing at Δ

_{1}+ Δ

_{2}+ Δ

*E*

_{70}

*(*

_{s}*R*)/

*ħ*= 0), respectively. When Δ

_{2}= 0 is fixed and

*R*is changed, the EIT window |00>−|10>−|20> is fixed at Δ

_{1}+ Δ

_{2}= 0 but the EIT window |00>−|10>−|40> moves. Especially, when

*R*= 80200 a.u. (Δ

_{4}+ Δ

*E*

_{70}

*(*

_{s}*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 |

*G*

_{2}

*G*

_{4+}0> and |

*G*

_{2}

*G*

_{4-}0>, as shown in Fig. 2(d5). When

*R*≠80200 a.u. (Δ

_{4}+ Δ

*E*

_{70}

*(*

_{s}*R*)/

*ħ*≠0) and Δ

_{1}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 (Δ

_{4}+ Δ

*E*

_{70}

*(*

_{s}*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 |

*G*

_{2+}

*G*

_{4+}0> and |

*G*

_{2+}

*G*

_{4-}0>, respectively (Fig. 2(a2)). Symmetrically, in the region withΔ

_{4}+ Δ

*E*

_{70}

*(*

_{s}*R*)/

*ħ*<0 where Δ

_{1}is scanned, the three peaks structure correspond to |

*G*

_{2+}0>, |

*G*

_{2-}

*G*

_{4+}0> and |

*G*

_{2-}

*G*

_{4-}0>, respectively.

The effect of Rydberg blockade can be also revealed when *R* is scanned at different Δ_{1}. In Fig. 2(b1), the baseline height of each curve at fixed Δ_{1} represents the probe transmission without the dressing effects of *E*_{4}, 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 Δ_{1} in Fig. 2(b2) represents the FWM signal without the dressing effects of *E*_{4}, 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 Δ_{1} 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 Δ_{1}, *R* can be always controlled to change in a region to guarantee that probe field has two-photon resonance or dressed state resonance. When Δ_{2} = 0 is fixed, and *R* is scanned with Δ_{1} = 0, on the one hand, the probe transmission shows pure-EIT (Fig. 2(b1)) at the two-photon resonance point Δ_{4} + Δ*E*_{70}* _{s}*(

*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

*E*_{1}could not resonate with |

*G*

_{2}

*G*

_{4 ±}0>, as shown in Fig. 2(d5), which weakens the wave mixing nonlinearity significantly. So we call the condition Δ

_{1}+ Δ

_{4}+ Δ

*E*

_{70}

*(*

_{s}*R*)/

*ħ*= 0 as

*suppression condition*. Second, with Δ

_{1}= −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

*E*_{1}first resonates with the dressed state |

*G*

_{2+}

*G*

_{4-}0> when

*R*is scanned from small to large (as a result Δ

*E*

_{70}

*(*

_{s}*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 |

*G*

_{2+}

*G*

_{4-}0> and that of undressed |10>, where ${{{\Delta}^{\prime}}^{\prime}}_{4}={\Delta}_{4}$ + Δ

*E*

_{70}

*(*

_{s}*R*)/

*ħ*–

*G*

_{2}represents the detuning of

*E*_{4}from |

*G*

_{2+}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 Δ

_{1}= −11 MHz, a pure-EIT of probe field and corresponding pure-suppression of ${E}_{F}^{DD}$ occur, as

*E*_{1}could not resonate with |

*G*

_{2+}

*G*

_{4 ±}0>, as shown in Fig. 2(d3). Fourth, when Δ

_{1}= −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

*E*_{1}resonates with |

*G*

_{2+}

*G*

_{4+}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 Δ

_{1}= −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

*E*_{1}could only resonant with |

*G*

_{2+}

*G*

_{4+}0>. When

*R*is scanned in the Δ

_{1}>0 region and the values of Δ

_{1}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 Δ

_{1}= 0, as shown in Fig. 2(b). It should be noted that the EIA of probe transmission or enhancement of FWM signal in the Δ

_{1}>0 region is due to

*E*_{1}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 Δ_{1} 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 Δ_{1} = 0, $-$11 and 11 MHz due to the interaction between *E*_{2} (${{E}^{\prime}}_{2}$) and *E*_{4}. The pure suppression at Δ_{1} = 0 is induced by the self-dressing effect, while pure suppressions at Δ_{1} = ± 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 Δ_{1} and Δ_{4}, Δ_{1} 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 *E*_{2} (${{E}^{\prime}}_{2}$) and *E*_{4}, 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 Δ_{4} 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 Δ*E*_{70}* _{s}*(

*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 *E*_{3} 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 Δ_{1} is scanned for a given *R*, the probe transmission always shows an EIT window on each curve (Fig. 3(a1)) at Δ_{1} + Δ_{4} + Δ*E*_{69}* _{d}*(

*R*)/

*ħ*= 0. As

*R*is changed from 58000 to 68300 a.u. (Δ

_{4}+ Δ

*E*

_{69}

*(*

_{d}*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

*E*_{1}resonates with |

*G*

_{4 ±}0> as shown in Fig. 3(a2), due to AT splitting.

Next, when *R* is scanned with fixed Δ_{1} and Δ_{3}, 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 Δ_{1} 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 *E*_{4}, 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 *E*_{4} participates in the generation of *E*_{S} directly, so *E*_{4} also offers self-dressing effect, that’s to say there is no external-dressing effect for SWM process. When *R* is scanned at Δ_{1} = 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 Δ_{1} + Δ_{4} + Δ*E*_{69}* _{d}*(

*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 Δ

_{1}<0 (Δ

_{1}= −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

*E*

_{1}first resonates with |

*G*

_{4+}0> (Fig. 3(d2)), i.e., the enhancement condition Δ

_{1}= −${{\Delta}^{\prime}}_{4}/2+\sqrt{{({{\Delta}^{\prime}}_{4}/2)}^{2}+|{G}_{4}{|}^{2}}$ is satisfied, with ${{\Delta}^{\prime}}_{4}$ = Δ

_{4}+ Δ

*E*

_{69}

*(*

_{d}*R*)/

*ħ*; and then two-photon resonance occurs, i.e., the suppression condition ${\Delta}_{1}\text{+}{{\Delta}^{\prime}}_{4}\text{=}0$ is satisfied. When Δ

_{1}changes to be positive (Δ

_{1}= 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

*E*_{1}resonates with the dressed state |

*G*

_{4-}0>, i.e., the enhancement condition Δ

_{1}= −${{\Delta}^{\prime}}_{4}/2-\sqrt{{({{\Delta}^{\prime}}_{4}/2)}^{2}+|{G}_{4}{|}^{2}}$ is satisfied, as shown in Fig. 3(d4). When Δ

_{1}is set far away from resonance point (Δ

_{1}= ± 80 MHz), the pure-EIA and corresponding pure-enhancement of

*E*_{S}are obtained because

*E*_{1}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 (Δ

_{1}−Δ

_{3}= 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 Δ_{1} and Δ_{4}, Δ_{1} and *R*, respectively. Similar to Figs. 2(c1)-(c2), avoided-crossing plots are obtained, in which the AT splitting due to dressing field *E*_{4} 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 ${E}_{S}^{D}$ versusΔ_{1} 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 Δ*E*_{69}* _{d}*(

*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, |40*s*40*s*> and |40*p*40*p*>, 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 |40*p*40*p*> (|40*s*40*s*>) is *E*(40*p*40*p*) + Δ*E*(40*p*40*p*) (*E*(40*s*40*s*) + Δ*E*(40*s*40*s*)), where *E*(40*p*40*p*) (*E*(40*s*40*s*)) is the eigen-energy without any blockade, and Δ*E*(40*p*40*p*) (Δ*E*(40*s*40*s*)) is the energy level shift due to primary blockade. So the energy gap between |40*s*40*s*> and |40*p*40*p*> without secondary blockade is Δ = *E*(40*p*40*p*)−*E*(40*s*40*s*) + Δ*E*(40*p*40*p*)−Δ*E*(40*s*40*s*), as shown in Fig. 4(a)
. Because *E*(40*p*40*p*) (*E*(40*s*40*s*)) can be controlled by the external electric field intensity *ε* due to a consequence of the enormous Stark shifts, and Δ*E*(40*p*40*p*) (Δ*E*(40*s*40*s*)) by *R*, the energy gap Δ*E* by both *ε* and *R*. When *ε* = 0 is fixed, the energy level shift curves versus *R* of |40*s*40*s*> and |40*p*40*p*> are plotted in Fig. 4(b), from which Δ*E*(*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 |40*p*40*p*> and |40*s*40*s*> is

*E*(40

*s*40

*s*) is set as zero. The eigen-energies are:

*Ε*

_{|}

_{ss}_{>+}(

*ε*,

*R*) and Δ

*Ε*

_{|}

_{ss}_{>−}(

*ε*,

*R*) are the energy level shift of |40

*s*40

*s*> after the primary and secondary Rydberg interactions, for the upper energy level and lower energy level, respectively. The energy level shifts are illustrated in Fig. 4(c), in which three points

*A*,

*B*and

*C*represent three important regimes of Rydberg interaction, which are classified according to the relative values of |Δ

*Ε*(

*ε*,

*R*)/2| and $\left|{\mu}_{1}^{2}/{R}^{3}\right|$, and can be controlled to transfer into each other by tuning

*ε*.

Firstly, if |Δ*Ε*(*ε*, *R*)/2|>>$\left|{\mu}_{1}^{2}/{R}^{3}\right|$, Eq. (6) yields into

*Ε*

_{|}

_{ss}_{>+}≈Δ

*Ε*(

*ε*,

*R*) and $\Delta {E}_{|}{{}_{ss}}_{>-}\approx -{\mu}_{1}^{4}/\left({R}^{6}\cdot \Delta E\right)$. This is van der Waals interaction energy, i.e., only the van der Waals interaction works in the neighborhood of point

*C*.

Then, when *ε* is tuned to make Δ*E*(*ε*, *R*) approach zero, i.e., in the case of |Δ*Ε*(*ε*,*R*)/2|<<$\left|{\mu}_{1}^{2}/{R}^{3}\right|$, the system exhibits a pair state resonance, and

*E*

_{±}∝1/

*R*

^{3}, which can well describe the interaction in the neighborhood of point

*A*. What’s more, in the case of Δ

*E*(

*ε*,

*R*) is comparable with $\left|2{\mu}_{1}^{2}/{R}^{3}\right|$, both van der Waals and resonant dipole-dipole interactions will work. Such regime corresponds to the neighborhood of point

*B*.

If *ε* is fixed and Δ*E*(*ε*, *R*) only changes with *R*. When *R* is fixed, Δ*E*(*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 Δ_{1} 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:

*R*-dependent forms for Δ

*Ε*

_{|}

_{ss}_{> ±}in the three regimes of primary and secondary interactions are different, as described in Eqs. (6)-(8). Figures 5(a1)-(a3) are the numerical results of ${E}_{F}^{DD}$ in the three cases by Eq. (9.1), while Figs. 5(b1)-(b3) are the results of ${E}_{S}^{D}$ by Eq. (9.2) without Γ

_{40}+

*i*(Δ

_{1}+ Δ

_{4}+ Δ

*Ε*

_{|}

_{ss}_{> ±}/

*ħ*) term (in order to emphasize the enhancement and suppression). Here, only the results with Δ

*Ε*

_{|}

_{ss}_{>+}are presented, because Δ

*Ε*

_{|}

_{ss}_{>−}will bring similar pattern. As shown in Fig. 4(c), Δ

*Ε*

_{|}

_{ss}_{>+}decreases with increasing

*R*. As a result, when

*R*is scanned from small to large at Δ

_{1}= −100 MHz, which is far away from Δ

_{1}= 0,

*E*_{1}can only resonant with |

*G*

_{2+}

*G*

_{4+}0>, which leads to the pure-enhancement in all the three curves in the first column in Fig. 5(a). Then at Δ

_{1}= −50 MHz,

*E*_{1}will first have two-photon resonance (Δ

_{1}+ Δ

_{4}+ Δ

*Ε*

_{|}

_{ss}_{> ±}/

*ħ*= 0) and then resonant with |

*G*

_{2+}

*G*

_{4+}0>, which leads to the first suppression and then enhancement in the curves in the second column in Fig. 5(a). With Δ

_{1}= −20 MHz,

*E*_{1}will only have two-photon resonance, which leads to the pure-suppression in the third column. With Δ

_{1}= −10 MHz,

*E*_{1}will first resonant with |

*G*

_{2+}

*G*

_{4-}0> and then have two-photon resonance, which leads to the first enhancement and then suppression in the fourth column in Fig. 5(a). With Δ

_{1}= 0,

*E*_{1}will only have two-photon resonance again, which leads to the pure-suppression in the fifth column. The sixth to ninth columns are cases with Δ

_{1}set opposite values to the first 4 columns with respect to Δ

_{1}= 0, which have similar behaviors with the former four columns. It is obvious that there are three symmetry centers, in which the pure-suppression is obtained. A two-peak structure in the global profile composed of the baseline of each curve (dashed line) can be also found in each row in Figs. 5(a1)-(a3), which is due to the self-dressing effect.

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 Δ*Ε*_{|}_{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 |*G*_{4+}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 (Δ*E*_{69}* _{d}*(

*R*)) increases while that in the quadratomic system (Δ

*Ε*

_{|}

_{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, Δ*E*(*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 Δ_{1} in each curve. With *ε* scanned from small to large, Δ*E*_{|}_{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 Δ_{1}<0 and Δ_{1}>0 are different. With Δ_{1}<0, the enhancement condition is ${\Delta}_{1}+{\lambda}_{G}{{}_{{{}_{2}}_{+}}}_{{G}_{4\pm}}=0$, with ${\lambda}_{G}{{}_{{{}_{2}}_{+}}}_{{G}_{4\pm}}={{\Delta}^{\u2033}}_{4}/2\pm \sqrt{{\left(\Delta {\u2033}_{4}/2\right)}^{2}+{\left|{G}_{4}\right|}^{2}}+{G}_{2}$ being the eigen-frequency difference between |*G*_{2+}*G*_{4 ±} 0> and |10>, where ${{{\Delta}^{\prime}}^{\prime}}_{4}={\Delta}_{4}$ + Δ*E*_{|}_{ss}_{> ±} (*ε*)/*ħ*–*G*_{2} (Δ*E*_{|}_{ss}_{>+}(*ε*) for Fig. 6(a1) and Δ*E*_{|}_{ss}_{>-}(*ε*) for Fig. 6(a2)); With Δ_{1}>0, the enhancement condition is ${\Delta}_{1}+{\lambda}_{G}{{}_{{{}_{2}}_{-}}}_{{G}_{4\pm}}=0$, with ${\lambda}_{G}{{}_{{{}_{2}}_{-}}}_{{G}_{4\pm}}={{\Delta}^{\u2033}}_{4}/2\pm \sqrt{{\left(\Delta {\u2033}_{4}/2\right)}^{2}+{\left|{G}_{4}\right|}^{2}}-{G}_{2}$ with ${{{\Delta}^{\prime}}^{\prime}}_{4}={\Delta}_{4}$ + Δ*E*_{|}_{ss}_{> ±} (*ε*)/*ħ* + *G*_{2}. The suppression condition is always Δ_{1} + Δ_{4} + Δ*E*_{|}_{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).

Figures 6(b1) and 6(b2) are plotted by substituting Δ*E*_{|}_{ss}_{> ±} (*ε*) into Eq. (9.2) without the term Γ_{40} + *i*(Δ_{1} + Δ_{4} + Δ*E*_{|}_{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 Δ_{1}<0 and ${\Delta}_{1}+{\lambda}_{{G}_{4}{}_{-}}\text{=}0$ for Δ_{1}>0, where ${\lambda}_{{G}_{\text{4}\pm}}={{\Delta}^{\prime}}_{4}/2\pm \sqrt{{({{\Delta}^{\prime}}_{4}/2)}^{2}+|{G}_{4}{|}^{2}}$ with ${{\Delta}^{\prime}}_{4}={\Delta}_{4}$ + Δ*E*_{|}_{ss}_{> ±} (*ε*)/*ħ*. The suppression condition is always ${\Delta}_{1}+{{\Delta}^{\prime}}_{4}=0$, with Δ*E*_{|}_{ss}_{>+}(*ε*) for Fig. 6(b1) and Δ*E*_{|}_{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 Δ_{1} 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 Δ*E*_{|}_{ss}_{>+}(*ε*), i.e., the upper curve in Fig. 4(d), in which the decreasing rate of Δ*E*_{|}_{ss}_{>+}(*ε*) becomes slowed down with increasing *ε*. On the other hand, in a curve with a certain Δ_{1}, compared with the curve at its left side with smaller Δ_{1}, *ε* must be scanned in a region where smaller Δ*E*_{|}_{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 Δ*E*_{|}_{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 ${{E}^{\prime}}_{4}$ (*ω*_{4},${{k}^{\prime}}_{4}$,${{G}^{\prime}}_{4}$ and Δ_{4}) 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 70*S*_{1/2} and only the primary Rydberg blockade in the diatomic system is considered, the energy level shift of |40> could be written as Δ*E*_{70}* _{s}*(

*R*), and correspondingly the term

*d*

_{40}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*= 10

^{6}a.u., Δ

*E*

_{70}

*(*

_{s}*R*) is so small that it can be neglected. In such case, with a certain Δ

_{4}, the signal ${{E}^{\prime}}_{F}$ will have an emission peak at Δ

_{1}+ Δ

_{4}= 0, i.e., the position where

*E*_{1}and

*E*_{4}have two-photon resonance (Fig. 7(b1)), as shown in the top curve in Fig. 7(a1). When

*R*decreases, Δ

*E*

_{70}

*(*

_{s}*R*) becomes noticeable. As a result, at the position Δ

_{1}+ Δ

_{4}= 0, the two-photon resonance will occur at the position Δ

_{1}+ Δ

_{4}+ Δ

*E*

_{70}

*(*

_{s}*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 Δ

*E*

_{70}

*(*

_{s}*R*), which reflects the blockade effect. Next, as shown in Fig. 7(b3), we add a strong field

*E*_{5}(

*ω*

_{5},

*k*_{5},

*G*

_{5}and Δ

_{5}) 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

*E*_{4}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 Δ

_{1}+ Δ

_{4}= 0, i.e., the blockade effect is eliminated with Δ

_{5}= 0, −13, −22 and −42 MHz, respectively. It is because that Δ

_{5}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 Δ

_{1}+ Δ

_{4}= 0, in the second to fifth curves from top to bottom, with different dressing field Rabi frequencies

*G*

_{5}= 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

*G*

_{5}, which can be controlled by changing the power of

*E*_{5}in experiment.

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 Δ*E*_{1}(*R*), Δ*E*_{2}(*R*) and Δ*E*_{4}(*R*), respectively. For experimental demonstration, the configuration requires *E*_{2} and *E*_{4} 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 Δ*E*_{1}(*R*)/*ħ*, Δ*E*_{2}(*R*)/*ħ* and Δ*E*_{4}(*R*)/*ħ* can be obtained after substituting the real parameters into the *R-*dependent expressions of Δ*E*_{40}* _{p}*(

*R*), Δ

*E*

_{60}

*(*

_{s}*R*) and Δ

*E*

_{70}

*(*

_{s}*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 Δ

_{1}are plotted in Figs. 8(a) and 8(b), respectively.

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 *E*_{4}. The reason for the first EIA of probe transmission and corresponding peak of FWM signal curve is that *E*_{1} 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 *E*_{1} resonates with *|G*_{4+}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., Δ_{1} + Δ_{2} + Δ*E*_{2}(*R*)/*ħ* = 0, as shown in Fig. 8(c2). With Δ_{1} 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 Δ*E*_{40}* _{p}*(

*R*), Δ

*E*

_{60}

*(*

_{s}*R*) and Δ

*E*

_{70}

*(*

_{s}*R*) must be smaller to ensure

*E*_{1}with larger Δ

_{1}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.

## Acknowledgments

This work was supported by the 973 Program (2012CB921804), NNSFC (11104216, 10974151, 61078002, 61078020, 11104214, 61108017, 61205112), NCET (08-0431), RFDP (20110201110006, 20110201120005, 20100201120031).

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