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
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  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  have been experimentally demonstrated. Recently, it is encouraging that many atomic coherence phenomena, such as coherent population trapping , stimulated Raman adiabatic passage , EIT  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  and thermal vapor , 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 5S1/2 (F = 3, |0>), 5S1/2 (F = 2, |3>), 5P1/2 (|1>), 5D3/2 (|2>), and 70S1/2 (or 69D3/2, |4>) of 85Rb. The resonant wavelengths (frequencies) in this system are 795 nm (Ω1≈Ω3 = 377.1 THz) from both of 5S1/2 (F = 3) and 5S1/2 (F = 2) to 5P1/2, 762.1 nm (Ω2 = 393.3 THz) for 5P1/2 to 5D3/2, and 474.2 nm (Ω4 = 632.18 THz) for 5P1/2 to 70S1/2, respectively. If |4> is 69D3/2, 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 (ω2, , and Δ2) drive |1>−|2>; E3 (ω3, k3, G3 and Δ3) and (ω3, , and Δ3) connect |1>−|3>; E4 (ω4, k4, G4 and Δ4) drives |1>−|4>. The Rabi frequency Gi is defined as Gi = μijEi/ћ, where μij is the transition dipole moment between |i> and |j>. The frequency detuning Δi is defined as Δi = Ω i −ωi.
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 are on, a FWM signal EF dressed by strong beam E4 satisfying the phase-matching condition kF = k1 + k2− can be generated in the |0>−|1>−|2> subsystem. While if all the beams except E2 and are on, a SWM signal ES satisfying kS = k1 + k3− + 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 are on, the FWM process (denoted as F1) generating EF 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) , we can obtain the undressed third-order density element for EF, with , d10 = Γ10 + iΔ1, d20 = Γ20 + i(Δ1 + Δ2) and Γij being the transverse relaxation rate between |i> and |j>. Then, when all the beams except E2 and are on, the SWM process S1 generating ES happens in the |00>−|10>−|40> ladder-type EIT window. By the undressed perturbation chain (II) , we can obtain the undressed fifth-order density element for ES, with and d30 = Γ30 + i(Δ1−Δ3). Even though EF and ES propagate in approximately the same direction, we can separate them in the experiment by selecting different EIT windows.
When E1, E2, and E4 are all on, EF will be dressed and we rewrite it as . 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 |G4 ± >, and the corresponding diatomic level |10> will split into |G4 ± 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) , and we can obtain , 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 are strong sufficiently, they will also bring dressing effect, that |10> will be first split into |G2 ± 0> by E2 (), and then split into third energy levels |G2+G4 ± 0> or |G2-G4 ± 0>. As a result, the dressed perturbation chain responsible for the FWM process F1 generating the doubly-dressed signal changes into (IV) , and 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, ΔEns(R)≈−C6/R6−C8/R8−C10/R10 is for ns-ns interaction, ΔEnp(R)≈−C5/R5−C6/R6−C8/R8 for np-np interaction, and ΔEnp(R)≈−C5/R5−C6/R6−C7/R7 for nd-nd interaction . 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 70S1/2, and the corresponding energy shift of level |40> in the diatomic system can be written as ΔE70s(R). With this energy level shift, the resonant frequency Ω4 will change into . As a result, the modified frequency detuning can be written as . Therefore, the matrix density element responsible for the singly-dressed FWM is modified into with . And for doubly-dressed FWM, we haveFigs. 1(f) and 1(g), respectively.
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 + ΔE70s(R)/ħ is around |G2+0>, two secondarily split levels |G2+G4 ± 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 + ΔE70s(R)/ħ is around |G2-0>, |G2-G4 ± 0> could be obtained as shown in Figs. 2(d6)-(d9). When R is adjusted to make Δ4 + ΔE70s(R)/ħ resonant with the midpoint of |G2 ± 0>, the first and secondary dressing effects can together split |10> into |G2G4 ± 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 + ΔE70s(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 + ΔE70s(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 + ΔE70s(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 |G2G4-0>, as shown in Fig. 2(d5). When R≠80200 a.u. (Δ4 + ΔE70s(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 |G2 ± 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 + ΔE70s(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+G4-0>, respectively (Fig. 2(a2)). Symmetrically, in the region withΔ4 + ΔE70s(R)/ħ<0 where Δ1 is scanned, the three peaks structure correspond to |G2+0>, |G2-G4+0> and |G2-G4-0>, 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 , 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 + ΔE70s(R)/ħ = 0. On the other hand, a pure-suppression dip of is correspondingly obtained (Fig. 2(b2)) because the probe field E1 could not resonate with |G2G4 ± 0>, as shown in Fig. 2(d5), which weakens the wave mixing nonlinearity significantly. So we call the condition Δ1 + Δ4 + ΔE70s(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, is first enhanced and next suppressed (Fig. 2(b2)). The reason for the first EIA and corresponding enhancement of is that E1 first resonates with the dressed state |G2+G4-0> when R is scanned from small to large (as a result ΔE70s(R) of |40> changes from large to small), which enhances the probe absorption and nonlinearity. Thus, we call as enhancement condition, in which is the difference between the eigen-frequency of |G2+G4-0> and that of undressed |10>, where + ΔE70s(R)/ħ–G2 represents the detuning of E4 from |G2+0>. While the reason for the next EIT and corresponding suppression of 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 occur, as E1 could not resonate with |G2+G4 ± 0>, as shown in Fig. 2(d3). Fourth, when Δ1 = −25 MHz, the probe transmission shows first EIT with corresponding suppression of because the suppression condition is satisfied. While the reason for the following EIA in probe transmission and corresponding enhancement of is that E1 resonates with |G2+G4+0> and the enhancement condition is satisfied, with . 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 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 resonance to |G2-G4 ± 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 () 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 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 () 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 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 ΔE70s(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 + ΔE69d(R)/ħ = 0. As R is changed from 58000 to 68300 a.u. (Δ4 + ΔE69d(R)/ħ correspondingly changes from 70 to 70 MHz), the EIT window |00>−|10>−|40> moves from right to left. Meanwhile, presents double peaks at the positions where E1 resonates with |G4 ± 0> as shown in Fig. 3(a2), due to AT splitting.
Next, when R is scanned with fixed Δ1 and Δ3, as shown in Fig. 3(b3), the behavior of 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 and in Eq. (4), respectively. To emphasize the interaction between the light field dressing effect and Rydberg blockade, we also calculate by eliminating the term . 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 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 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 |G4 ± 0>, as shown in Fig. 3(d1)-(d5). We should note here that E4 participates in the generation of ES 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 + ΔE69d(R)/ħ = 0, where probe field could not resonate with either of the two dressed energy levels |G4 ± 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 in Fig. 3(b2), is that E1 first resonates with |G4+0> (Fig. 3(d2)), i.e., the enhancement condition Δ1 = − is satisfied, with = Δ4 + ΔE69d(R)/ħ; and then two-photon resonance occurs, i.e., the suppression condition 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 is that two-photon resonance first occurs; and then E1 resonates with the dressed state |G4-0>, i.e., the enhancement condition Δ1 = − 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 ES are obtained because E1 can only resonate with one of |G4 ± 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 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 is obtained along the vertical axis. Figure 3(c3) presents the intensity of 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 ΔE69d(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> isFig. 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 |ΔΕ(ε, R)/2| and , and can be controlled to transfer into each other by tuning ε.
Firstly, if |ΔΕ(ε, R)/2|>>, 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|<<, 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 and 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:Eqs. (6)-(8). Figures 5(a1)-(a3) are the numerical results of in the three cases by Eq. (9.1), while Figs. 5(b1)-(b3) are the results of by Eq. (9.2) without Γ40 + i(Δ1 + Δ4 + ΔΕ|ss> ± /ħ) term (in order to emphasize the enhancement and suppression). Here, only the results with ΔΕ|ss>+ are presented, because ΔΕ|ss>− will bring similar pattern. As shown in Fig. 4(c), ΔΕ|ss>+ decreases with increasing R. As a result, when R is scanned from small to large at Δ1 = −100 MHz, which is far away from Δ1 = 0, E1 can only resonant with |G2+G4+0>, which leads to the pure-enhancement in all the three curves in the first column in Fig. 5(a). Then at Δ1 = −50 MHz, E1 will first have two-photon resonance (Δ1 + Δ4 + ΔΕ|ss> ± /ħ = 0) and then resonant with |G2+G4+0>, which leads to the first suppression and then enhancement in the curves in the second column in Fig. 5(a). With Δ1 = −20 MHz, E1 will only have two-photon resonance, which leads to the pure-suppression in the third column. With Δ1 = −10 MHz, E1 will first resonant with |G2+G4-0> and then have two-photon resonance, which leads to the first enhancement and then suppression in the fourth column in Fig. 5(a). With Δ1 = 0, E1 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 Δ1 set opposite values to the first 4 columns with respect to Δ1 = 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 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 |G4-0>. 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 (ΔE69d(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 , with being the eigen-frequency difference between |G2+G4 ± 0> and |10>, where + 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 , with with + 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).
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 for Δ1<0 and for Δ1>0, where with + ΔE|ss> ± (ε)/ħ. The suppression condition is always , 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 in the three-level system composed of |00>, |10> and |40>, as shown in Fig. 7 (b1), which includes an additional laser beam (ω4,, and Δ4) connecting |40>−|10>. Compared with the system in Fig. 1(a), the density matrix element is . If |4> is set as 70S1/2 and only the primary Rydberg blockade in the diatomic system is considered, the energy level shift of |40> could be written as ΔE70s(R), and correspondingly the term d40 will change into . When R = 106 a.u., ΔE70s(R) is so small that it can be neglected. In such case, with a certain Δ4, the signal 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, ΔE70s(R) becomes noticeable. As a result, at the position Δ1 + Δ4 = 0, the two-photon resonance will occur at the position Δ1 + Δ4 + ΔE70s(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 ΔE70s(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 |G5 ± 0> with the eigen-frequencies as. So the detuning of E4 is modified into . As a result, two peaks occur in Figs. 7(a2) and 7(a3), when the two-photon resonance conditions 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 . 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 is satisfied in each curve due to the adoption of appropriate G5, which can be controlled by changing the power of E5 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 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 could be written as
In calculation, we set |1>, |2> and |4> as 40P1/2, 60S1/2 and 70S1/2, 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 ΔE40p(R), ΔE60s(R) and ΔE70s(R), respectively. They all present a declining trend as R increases. The calculated intensity of and probe transmission versus R at different Δ1 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 and the peaks at two sides correspond to .
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 |G4 ± 0> by E4. The reason for the first EIA of probe transmission and corresponding peak of FWM signal curve is that E1 resonates with |G4-0> (Fig. 8(c1)), thus the enhancement condition is satisfied with being the frequency detuning between |10> and |G4-0>, in which . 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 is satisfied with . 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 ΔE40p(R), ΔE60s(R) and ΔE70s(R) must be smaller to ensure E1 with larger Δ1 get two-photon and dressed state resonances.
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.
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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