Phase Modulation in Rydberg Dressed Multi-Wave Mixing processes

We study the enhancement and suppression of different multi-waving mixing (MWM) processes in a Rydberg-EIT rubidium vapor system both theoretically and experimentally. The nonlinear dispersion property of hot rubidium atoms is modulated by the Rydberg-Rydberg interaction, which can result in a nonlinear phase shift of the relative phase between dark and bright states. Such Rydberg-induced nonlinear phase shift can be quantitatively estimated by the lineshape asymmetry in the enhancedand suppressed MWM processes, which can also demonstrate the cooperative atom-light interaction caused by Rydberg blockaded regime. Current study on phase shift is applicable to phase-sensitive detection and the study of strong Rydberg-Rydberg interaction.

The interaction among Rydberg atoms scales with n 11 and leads to the change in refractive index of the medium and nonlinear phase shift of the relative phase between dark and bright states, which can be mapped onto the enhancement and suppression of EIT-assisted MWM processes with dressing effects. To be specific, the modification of refractive index (n r ) caused by Rydberg energy level shift 22 (Δ ω 2 ) can be expressed as ω ω ∆ = (∂ /∂ )∆ , ( ) n n 1 r r 2 2 where ∂n r /∂ω 2 = (n g − 1)/ω 2 , ω 2 is the Rydberg state coupling laser frequency and n g is the group refractive index. The theoretical simulation of Δ n r is shown in Fig. 1(c). The phase modulation (Δ Φ 1 ) due to the strong cooperative atom-light interaction due to Rydberg blockade is described as where N(v) = N 0 exp(− v 2 /u 2 )/uπ 1/2 is the particle number density in terms of speed distribution function 15 ; Ω i = d ij E ij /ħ (i,j = 1, 2…) is the Rabi frequency between |i〉 ´|j〉 , and d ij is the dipole momentum; N 0 is the atom density; γ 1 = (Γ 10  and   ρ  ρ  ρ  ρ  ρ   00  0  10  1  30  2  10  3  40  4   1  3  3  4 , the corresponding fifth-order polarizations P (5)  .
Here, the additional phase factors e iΔΦ and e iΔΦ′ are introduced into the dressing terms (|Ω 2 |/n 11 ) 0.4 /γ 2 and |Ω 4 | 2 /γ 4 to account for the propagation effect. Δ Φ = Δ Φ 1 + Δ Φ 2 , where Δ Φ 1 (U) is the phase modulation induced by the possibly coherent Rydberg-Rydberg interaction U; the relative phase Δ Φ 2 and Δ Φ ′ are related to the orientations of induced dipole moments and can be manipulated 18 by corresponding laser frequency detuning and Rabi frequency. Phase modulated intensity and spatial effects in the Y-type subsystem. Figure 2 shows the dressed FWM1 process in the Y-type four-level subsystem |0〉 ↔|1〉 ↔|2〉 ↔|4〉 by scanning the frequency of Rydberg coupling field E 2 . Suppressed and enhanced FWM1 signals (the suppressed condition is Δ 1 + Δ 4 = 0 and the enhanced condition is ∆ ∆ are observed by changing the frequency detuning of E 1 or E 4 . According to the new two-photon dressed rule 26 , the moving states |± 〉 will impose influence on the enhancing and suppressing results of MWM signals. Let's first show the generating process of Rydberg-dressing enhancement and suppression simply. Figure 2(a) shows the switch from an enhanced peak to a suppressed dip by growing Ω 1 at Δ 1 = − Δ 4 = 30 MHz. The dressing processes can be considered as following: first, as shown in Fig. 2(f), level |1〉 is split into the dressed states |± 1 〉 by E 1 ; and then |+ 1 〉 is split into |+ 1 ± 2 〉 secondly by E 2 . Therefore, once the dressing level |+ 1 〉 moved around the position of Δ 1 , the suppressed condition is satisfied and the suppressed case of FWM1 occurs in Fig. 2(a). Figure 2(b) shows the dependence of suppressed dip on the strength of E 4 at Δ 1 = Δ 4 = 0. The enhanced condition cannot be satisfied in the situation shown in Fig. 2(b) in which the suppressed dip increases as the power of E 4 increases and the two-step dressing process can be simplified as level |1〉 is split into |± 1 〉 .
In order to visually investigate the nonlinear dispersion property induced by Rydberg dressing effect and cooperative effect, we turn to the spatial effects on the images of dressed signals. With only E 1 and E 2 turned on, Fig. 2(c) shows the focusing/defocusing effects of probe signal versus Δ 1 . Nonlinear refractive index n r is negative in the self-focusing medium (Δ 1 < 0) while positive in the self-defocusing one (Δ 1 > 0). Figure 2(d1,d2) show the probe images with E 1 &E 4 and E 1 &E 2 &E 4 on versus Δ 1 , respectively. With E 2 blocked, the focusing/defocusing effects of probe images at different Δ 1 + Δ 4 = 0 can be stronger than the effects in Fig. 2(c) due to the growing of absolute value of refractive index. With E 1 &E 2 &E 4 on, the images of dressed E 4 EIT become more defocusing compared with the corresponding ones in Fig. 2(d1) due to Δ n r is negative in most part of the resonance line as shown in Fig. 1(c). In addition, the spatial splitting and shift in Fig. 2 (a) Switching between enhanced peak to suppressed dip by increasing  Figure 2(e1,e2) are the images of dressed E 2 EIT and Rydberg dressed FWM1 versus Δ 2 , respectively. The dressed FWM1 and dressed E 2 EIT with Δ 1 = Δ 2 = Δ 4 = 0 are much more defocusing than the points of Δ 2 ≠0. All the signal images visually advocate the modulation on dispersion property due to the existence of Rydberg-Rydberg interaction. Figure 2(g) shows the change in dressed enhancement and suppression of FWM1 by increasing the frequency detuning Δ 1 at Δ 4 = 0. The Lorentzian profile (curve constituted of the baseline of each signal) is a one-photon peak of the FWM1 signal versus Δ 1 and can be described by the single-photon term γ 1 in Eq. (3). The intensity of FWM1 in Fig. 2(g) is first suppressed and then enhanced at Δ 1 = −32 MHz, while it is first enhanced and then suppressed at Δ 1 = 32 MHz. Obviously, a dressing asymmetry occurs with Δ 1 = 0 considered as a center.
In general, the dressing enhancement peaks and suppression dips are symmetrical distributed along the center. However, the induced nonlinear phase shift may lead to the asymmetry 18,21 in the lineshapes of dressed MWM signals. To estimate such dressing asymmetry quantitatively, we define the asymmetry factor as where e i and s i represent the enhancement and suppression of FWM1 intensity; subscripts 2 and 1 indicate e i (or s i ) are taken with Δ 1 > 0 and Δ 1 < 0, respectively. Actually, the relationship between A F and phase shift can be described as where Δ FWHM and β are the full width at half maximum (FWHM) and full width at a certain frequency detuning point of the corresponding profile, respectively; α 1 and α 2 are the ratio parameters for phase shift Δ Φ and Δ Φ ′ caused by E 2 and E 4 , respectively. According to Eq. (8), the value of A F in Fig. 2(g) is about 0.58 at |Δ 1 | = 32 MHz. Due to the absence of Autler-Townes (AT) splitting on the profile, the dressing effect of E 4 on the one-photon term γ 1 that only affects the intensities of the signals can be neglected. Since the modulated results of FWM1 in Fig. 2(g) are related to the change in Δ 1 , one can attribute the results to the dressing effect of E 2 on γ 1 . Therefore, A F in Fig. 2(g) is mainly contributed by the Rydberg dressing and cooperative nonlinear effect. The denominator of Eq. (3) is simplified to [γ 1 + (|Ω 2 | 2 /n 11 ) 0.4 e iΔΦ /γ 2 ] 2 γ 4 and can explain Fig. 2 Fig. 2(g1)). Figure 2(h) is the modulated enhancement and suppression of FWM1 signal by increasing Δ 4 at Δ 1 = 0, and A F is about 0.91 at |Δ 4 | = 50 MHz. Different from the case in Fig. 2(a), the Lorentzian profile (curve constituted of the baseline of each signal) is a two-photon peak of the FWM1 signal versus Δ 4 , which can be described by the two-photon term γ 4 in Eq. (3). Obviously, the change of Δ 4 can also affect the modulated results of FWM1, and it can be ascribed to the dressing effect of E 2 on γ 4 associating with selfdressing shown in Eq. (3). As a consequence, the denominator of Eq. (3) is simplified as Phase modulated intensity in the inverted-Y type subsystem. Now, we try to pick out the phase shift induced by the Rydberg blockade. Figure 3 shows the enhanced and suppressed FWM2 coexisting with the SWM2 by scanning Δ 4 at discrete Δ 1 . To be specific, Fig. 3(a) is the case with E 2 beam blocked and shows the dressing effect of E 4 on FWM2 versus Δ 4 at different Δ 1 , which can be well simulated by Eq. (4) by setting Δ Φ ′ = − π/6 at Δ 3 = 150 MHz (see Fig. 3(a1)). As defined above, the dressing asymmetry factor A F in Fig. 3(a) is 0.19 at |Δ 1 | = 80 MHz. The profile (curve constituted of the baseline of each signal) in Fig. 3(a) is the one-photon peak of FWM2 signal versus Δ 1 (see the one-photon term γ 1 in Eq. (4)) with E 2 blocked, and the peak is broadened to be 200 MHz by the Doppler effect Δ 1 − Δ 3 = k 1 v + k 3 v. Figures 3(b,c) are the ones with the dressing effect of E 2 (coupling the transition between 5P 3/2 ↔54D 5/2 ) at different atomic densities, respectively. The profiles in Figs.3 (b) and (c) are the peaks of FWM2 signal together with SWM2 signal by scanning Δ 1 . However, the dressed FWM2 signal is restrained in a narrower range by the EIT configuration of |0〉 ↔|1〉 ↔|4〉 . Compared with Fig.3(a), A F values in Fig. 3(b,c) increase to be as high as 0.61 and 0.86 at |Δ 1 | = 80 MHz due to the introducing of Rydberg field. The difference between the asymmetry factors on the profiles can be explained by the nonlinear phase shift caused by E 2 dressing effect and the cooperative atom-light interaction 27 . Since both Fig. 3(b,c) are related to the same Rydberg state 54D 5/2 , the phase shift induced by the change of cooperative nonlinearity due to Rydberg-Rydberg interaction can be observed by comparing the modulated results of N 0 = 1 × 10 12 cm −3 and N 0 = 2.4 × 10 12 cm −3 . The introducing of correlations between atoms into atom-light interaction can lead to a cooperative effect. The increase of Rydberg atom population will increase the cooperative nonlinearity and result in a dramatically change of the measured lineshapes.
Comparing the fourth curve in Fig. 3(b) with the fourth one at Δ 1 = 80 MHz in Fig. 3(a), the difference between the modulated results can be explained well by setting Δ Φ = Δ Φ 1 + Δ Φ 2 = − π/12 (see Fig. 3(b1)). For the higher density shown in Fig. 3(c), the theoretical prediction agrees well with the experimental results by setting Δ Φ = Δ Φ 1 + Δ Φ 2 = − π/3 (see Fig. 3(c1)). Obviously, the phase shift as well as the dressing asymmetry factor grows with the atomic density and such density-dependent characteristic can demonstrate the Δ Φ 1 caused by the change of cooperative nonlinearity. Considering that the values of Δ Φ 2 in Fig. 3(b,c) are almost same due to the saturated dressing effect, the phase difference caused by the increase of cooperative nonlinear effect is approximately π/4. Therefore, such results sufficiently prove the existence of the phase shift induced by the interaction between Rydberg atoms.
Besides of the blockade dressed SWM process discussed above, one can further use the Rydberg MWM process to study the phase shift induced by the strong Rydberg-Rydberg interaction. Figures 4(a,b) show the induced enhancements and suppressions of FWM2 and SWM1 together with the SWM2 processes for 37D and 54D by varying Δ 1 at Δ 2 = Δ 3 = 0, respectively. The peaks of the FWM2 and SWM1 signals versus Δ 1 for 37D and 54D are shown by the Lorentzian profiles, which can be described by the one-photon term γ 1 in Eqs. (4) and (5), respectively. Since the results are related to the changing of Δ 1 , they can be attributed to the dressing effects on the one-photon term γ 1 as shown in Eqs. (4) and (5). The phase shift of Δ Φ ′ on the dressing term |Ω 4 | 2 /γ 4 is − π/6 (see Fig. 4(a1,b1)). The difference of the phase shifts induced by different cooperative nonlinear effect for the two principal quantum numbers can be obtained by comparing the corresponding modulated results at the same frequency detuning. In the current case, a phase shift difference of π/4 is introduced between 37D and 54D due to the n-dependent characteristic of cooperative nonlinearity. Figure 4(c,d) are the enhanced and suppressed SWM1 for 37D and 54D at Δ 1 = -Δ 3 = 30 by altering Δ 2 , respectively. The Lorentzian profiles with linewidth of 60 MHz are the two-photon peaks of the SWM1 signal versus Δ 2 for 37D and 54D, respectively, and related to the two-photon term γ 2 in Eq. (5) (see Fig. 4(c1,d1)). Different from the former cases, we are now interested in the dressing effects on the two-photon term γ 2 whereas the dressing effects on γ 1 can be neglected. However, except for the increase of suppression in correspondingly modulated SWM1 signals of 37D and 54D, the dressed results are almost the same for both states due to the strong optical pumping. Therefore, the information of the phase difference in inverted-Y subsystem with optical pumping effect by changing Δ 2 is not as  obvious as in Y-type system by changing Δ 4 . Here, we have to mention that the central frequency shift of the Lorentzian profiles is observed due to the energy shift induced by different Rydberg-Rydberg interactions.
Finally, we characterize the blockaded enhancement and suppression results at Δ 1 = -120 MHz in Fig. 4(a,b) as the functions of the probe field strength P 1 , the Rydberg state coupling field strength P 2 , and the coupling field strength P 4 for three nD 5/2 states. We expand Eqs.  2 , and Ω a , respectively. In addition, we have I∝ n −3 according to Rydberg dressed MWM intensity I∝ |Ω 2 | 2 ∝ |d ij | 2 and d ij ∝ n (−3/2) . Consequently, the Rydberg dressed signals for each principal quantum number n are scaled to n = 37 by the factor (n * /37 * ) 3 accounting to the decrease in d ij with increasing n. Here, n * = n− δ, and δ = 1.35 is due to the quantum defect for nD 5/2 state 27 . Figure 5(a) presents the E 1 power dependences of the (a1) enhanced peak, (a2) suppressed dip and (a3) background, respectively, for three nD 5/2 states. The change of enhanced peak is mainly contributed by the enhanced FWM2 & SWM1 processes and the two-photon peak of SWM2. The trend of the suppressed dip can be understood as the saturating dressing-effect of E 4 at Δ 1 + Δ 4 = 0. The background evolution is due to the sum of FWM2 and SWM1 processes. Based on the evolutions of enhancement and suppression, we can draw the conclusion that the dressing asymmetry A F increases with the strength of E 1 . The saturating dressing-effect of E 4 means the phase shift is mainly caused by the existence of E 2 dressing and blockaded effect.  The cases of the E 2 power dependences for three nD 5/2 states are shown in Fig. 5(b). First, we focus on the P 2 dependence of the enhanced peak (see Fig. 5(b1)). At the low excitation intensity, the enhanced FWM2 signal and SWM2 signal contribute to the enhanced peak. As Ω 2 increases, the enhanced SWM1 signal also makes the height of enhanced peak increase. However, the blockade term (|Ω 2 |/n 11 ) 0.4 makes the curve saturated at higher power level. Then, the descending part of the curve is due to the dressing effect of E 2 associated with its excitation blockade effect from |Ω 2 |/n 11 ) 0.4 e iΔΦ /γ 2 . Next, the power dependence of the suppressed dip can also reflect the blockade effect and the dressing effect of E 2 (see Fig. 5(b2)). Initially, the saturated dressing of E 4 on FWM2 signal at Δ 1 + Δ 4 = 0 and gradually increased SWM1 signal are the main factors. Then the curve becomes saturated due to the blockade term (|Ω 2 |/n 11 ) 0.4 at higher power level of E 2 . As the power further increasing, the interaction between two dressing processes weakens the dressing results. Finally, one can obtain the direct blockade effect from the P 2 power dependence of the background as shown in Fig. 5(b3). The background is consisted of FWM2 and SWM1 signals without the dressing effect of E 4 . The saturation is due to the blockade effect, and the descending part is due to the combination of blockade effect and dressing effect of E 2 . Given the above descriptions and analysis of peak and dip evolution corresponding to P 2 strength dependence, one can deduce that the phase modulation as well as asymmetry can become more obvious by strengthening E 2 . Meanwhile, we must note that the different principles for the increase of asymmetry are very corresponding to the three stages of power increase mentioned above. Lastly, E 4 power dependences in Fig. 5(c) just show the regular enhancement and suppression processes by E 4 . The asymmetry changes are mainly aroused from the dressing effect of E 4 .

Discussion
The dressed suppression and enhancement of blockade MWM processes can reveal the change in nonlinear refractive index induced by cooperative atom-light interactions and corresponding dressing effects in Rydberg-EIT hot medium. On one hand, the observation of spatial shift and splitting effects of corresponding signals can visually advocate the dispersion property change of medium under blockaded effect. The transverse wave vector to explain the spatial effects is defined as Figure 5. Power dependences (P 1 , P 2 and P 4 respectively) of the (a1, b1, c1) enhanced peaks, (a2, b2, c2) suppressed dips, and (a3, b3, c3) backgrounds, respectively, for three different nD 5/2 states. The intensities of the Rydberg signals are scaled by (n * /37 * ) 3 to account for the n dependence of the dipole matrix elements. The first-order differential δ ∆Φ ξ = ∂ /∂ ⊥ k 1 can describe spatial shift/splitting effects and the second-order differential ∆Φ ξ ∂ /∂ 2 1 2 can explain the focusing/defocusing effects. On the other hand, the intensity modification of the enhanced and suppressed MWM signals obtained by scanning the dressing fields, which essentially control dark and bright states, can reflect the change in refractive index of a medium for a laser or MWM signals. Further, the cooperative nonlinearity induced phase modulation can be proportional to the refractive index change caused by Rydberg energy level shift. Consequently, we can quantificationally map the phase shift by cooperative nonlinear interaction onto suppression and enhancement of MWM processes involving in Rydberg states. With the dressing asymmetry A F on the modulated results defined, A F ∝ (Δ FWHM /β )(α 1 Δ Φ + α 2 Δ Φ ′ ) is established to depict the phase shift between dressing dark and bright states, where Δ Φ includes the phase shifts from both Rydberg dressing states and Rydberg excitation blockade and Δ Φ ′ results from the orientations of induced dipole moments. The parameters α 1 and α 2 can be determined by experimental parameters such as the frequency detunings, Rabi frequencies, atom density and polarization states of laser fields.

Methods
Experimental setup. We use six light beams from three commercial external cavity diode lasers (ECDL) and one frequency-doubling laser system to couple a five-level X-type rubidium atomic system. The transition of D 2 line is driven by weak laser beam E 1 stabilized to a temperature-controlled Fabry-Perot (FP) cavity. A pair of coupling beams E 3 and E 3 ′ , also driving the transition of D 2 line for different hyperfine configuration, are from another ECDL locked to the saturated absorption signal of rubidium atom. Beam E 2 driving the Rydberg excitation is a frequency-doubled laser with high stability. We get the needed 480 nm laser E 2 by the way of frequency doubling LD2 at ~960 nm with a periodically-poled KTP crystal in an external ring resonator to generate the second harmonic wave. The strong beam E 2 adding onto the beam E 3 (in the same direction), which counter-propagates with beam E 1 , drives the highly-excited Rydberg transition. E 4 and E 4 ′ are from the same LD4. E 4 adds onto the beam E 3 by a cubic polarizing beam splitter (PBS) and E 4 ′ propagates with E 3 ′ symmetrically with respect to E 2 . All beams are focused by two lenses (L1 and L2, respectively) with same focal length 500 mm before the cell and intersect at one point inside the cell. The 1 cm long rubidium cell is wrapped by μ-metal and heated by the heater tape. The optical depth (OD) is 70 for atom density of 1.0 × 10 12 cm −3 .
Theoretical models for Δn r and ΔΦ 1 (U). Nonlinear refractive index change is modeled by taking Δ n r as the product of the slope of the dispersion (∂n r /∂ω 2 ) and the energy level shift (Δ ω 2 ) of the Rydberg state due to the Rydberg-Rydberg interaction. ∂n r /∂ω 2 is derived from the real part of the complex susceptibility 28 χ for stationary atoms and zero-coupling detuning as where U(r − r′ ) is the cooperative nonlinear interaction for Rydberg atoms at nD states; N 2 is the density of excited Rydberg atoms. If we calculate the Rydberg excitation density via optical Bloch equation (OBE) by using the mean-field model 2 and taking 3 into account, the average Rydberg atom density ρ e with considering of Doppler width Ω D can be described as Here R d is the radius of a Rydberg domain, which includes a single Rydberg atom and many ground-state atoms. By comparing with the non-blockade case, we find the following regulation as  →  . , where γ 31 = Γ 13 + iΔ 3 ; C is a constant mainly determined by the coefficient of Rydberg-Rydberg interaction and resulting from numerical integration outside the given sphere and the atom excitation efficiency between |0〉 and |1〉 . Therefore, the change in refractive index can be defined as