Phase Modulation in Rydberg Dressed Multi-Wave Mixing processes

Abstract

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.

Introduction

The phase modulation as well as the refractive index modification in a Rydberg medium, caused by electric fields produced either externally or internally owing to the interparticle interactions, is of central importance in nonlinear optics, laser technology, quantum optics and optical communications¹. Because the high-lying Rydberg electron is very far from the core of the atom, the atom possesses exaggerated properties, such as huge polarizability that scales as n⁷, where n is the principle quantum number. These properties lead to strong and tunable Rydberg-Rydberg interactions^2,3,4 among the atoms, which can render the Rydberg medium intrinsically nonlinear. For example, Rydberg electromagnetically induced transparency (EIT) makes the transmission through the medium highly sensitive to electric fields¹, which can enable modifications on the refractive index and nonlinear phase shift due to the interparticle interactions in the nonlinear processes associated with EIT.

Comparing with the other nonlinear optical processes, the multi-waving mixing (MWM) processes in Rydberg-EIT medium have unique features^5,6,7,8 and one typical feature is that the coherence time of the generated signal is shorter than the time of ionization⁹, while it is known that the incoherence plasma formation in Rydberg gases is ~100 ns or longer^10,11. With EIT configuration, the coherence between the ground state and highly-excited Rydberg states is well established, which can enhance the efficiency of the MWM processes^12,13. In addition, the spatial arrangement of EIT configuration will suppress the Doppler width greatly, which makes atoms in the beam volume behave like cold atoms with reduced Doppler effect^14,15,16,17. Finally, the EIT windows will pick up the corresponding MWM signals with narrow linewidth (less than 30 MHz). Therefore, probing the EIT-assisted MWM processes can provide a powerful spectral method to investigate the properties of Rydberg atoms.

In this paper, we study the enhancement and suppression of Rydberg dressed MWM processes with the assistance of EIT windows in a hot Rb atomic system both theoretically and experimentally. The enhanced and suppressed MWM signals are significantly modified via the relative phase control¹⁸ due to the nonlinear dispersion property modification induced by corresponding dressing effects and the cooperative nonlinear effect^19,20 from the Rydberg blockade regime. The introducing of strong Rydberg-Rydberg interactions into atom-light interaction means that each atom can no longer be treated independently and the correlations between atoms must be taken into consideration, which can be interpreted as a cooperative effect. The cooperative nonlinearity in an atomic ensemble can be much more obvious for high atomic density. As s result, the spatial effects of corresponding dressed signals can visually advocate the change of nonlinear dispersion property in current experiment. The intensity evolutions of enhancement and suppression results may map onto the nonlinear phase shift in modulated dispersion property by scanning dressing fields. Different from the asymmetry degree of cavity transmission profile method²¹, such nonlinear phase shift with background-free advantages can be estimated via the dressing asymmetry in enhanced and suppressed MWM linshapes, which can also demonstrate the excitation blockade effects. The nonlinear phase shift of the relative phase between dark and bright states gives a novel way for studying the Rydberg-Rydberg interactions and phase-sensitive detection.

Results

An X-type five-level ⁸⁵Rb atomic system, consisting of two hyperfine states F = 3 (|0〉) and F = 2 (|3〉) of the ground state 5S_1/2, a first excited state 5P_3/2 (|1〉), a lower-lying excited state 5D_3/2 (|4〉) and a highly-excited Rydberg state nD_5/2 (|2〉), is used to generate the EIT-assisted MWM processes. Six laser beams derived from four commercial external cavity diode laser systems with frequency-stabilized servos are coupled into the corresponding transitions as shown in Fig. 1(a). The experimental setup is shown in Fig. 1(b). Except for the E₄′, the experimental setup is essentially the same as previous work²². A weak laser beam E₁ (780.24 nm with a diameter of 0.8 mm, frequency ω₁, wavevector k₁) from LD1 probes the lower transition |0〉 to |1〉, while a pair of coupling beams E₃ (780.23 nm, ω₃, k₃) and E₃′ (ω₃, k₃′), derived from the same LD3 with a small angle between them both with the same diameter of 1 mm, connect another lower transition |3〉 to |1〉. To excite hot rubidium atoms from level |1〉 to Rydberg states |2〉, we obtain the needed 480 nm laser E₂ (ω₂, k₂) by the way of frequency doubling LD2 at ≈960 nm. The strong beam E₂ (diameter 1 mm) adding onto the beam E₃ (in the same direction), which counter-propagates with beam E₁, drives the highly-excited Rydberg transition |1〉 to |2〉. E₄ (775.98 nm with a diameter of 1 mm, frequency ω₄, wavevector k₄) and E₄′ (ω₄, k₄′) from LD4 drive the transition |1〉 to |4〉.

Figure 1

(a) A five-level atomic system in Rydberg-EIT rubidium atom for dressed MWM processes. (b) Experimental setup for different MWM processes. L-lens, D-detector, FD-frequency doubler, HW-half wave plate with corresponding wavelength, PBS-polarized beam splitter with corresponding wavelength. Double-headed arrows and filled dots denote horizontal polarization and vertical polarization of the incident beams, respectively. (c) Theoretical calculations corresponding to the change in refractive index (Δn_r) of a medium for a probe laser (or MWM signal) frequency versus Δ₁ and Δ₂. Ω₁ = 2π × 54 MHz, Ω₂ = 2π × 7.6 MHz, Ω₄ = 2π × 142 MHz, Ω₄′ = 2π × 224 MHz. The atom density is 1.0 × 10¹² cm⁻³.

Different-order dressed MWM processes can be obtained by turning the incident beams on selectively. First, by blocking beams E₃ and E₃′, a four-wave mixing (FWM) process E_FWM1 with the phase-matching condition (PMC) k_FWM1 = k₁ + k₄ − k₄′ can be dressed by E₂ in the Y-type four-level subsystem |0〉↔|1〉↔|2〉↔|4〉. Next, when opening all other beams except E₄′, a non-EIT-assisted FWM process E_FWM2 (with k_FWM2 = k₁ + k₃ − k₃′ in the Λ-type three-level subsystem |0〉↔|1〉↔|3〉) and two EIT-assisted six-wave mixing (SWM) proesses^14,23,24 E_SWM1 involving in Rydberg states and E_SWM2 (with the PMCs of k_SWM1 = k₁ + k₃ − k₃′ + k₂ − k₂ and k_SWM2 = k₁ + k₃ − k₃′ + k₄ − k₄) can be observed in |0〉↔|1〉↔|3〉↔|2〉 and |0〉↔|1〉↔|3〉↔|4〉, respectively. These MWM signals have the same emitting direction (opposite to the direction of E₃′, as shown in Fig. 1(b)) except for E_FWM1 (propagating along the opposite direction of E₄′). The various MWM processes are identified by tuning the frequency detuning of corresponding coupling beams and detected by respective avalanche photodiode detectors (APD). Specifically, the MWM processes related to the Rydberg state |2〉 may be called Rydberg MWM signals with strong Rydberg-Rydberg interactions.

The interaction among Rydberg atoms scales with n¹¹ 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 shift²² (Δω₂) can be expressed as

where ∂n_r/∂ω₂ = (n_g − 1)/ω₂, ω₂ is the Rydberg state coupling laser frequency and n_g is the group refractive index. The theoretical simulation of Δn_r is shown in Fig. 1(c). The phase modulation (ΔΦ₁) due to the strong cooperative atom-light interaction due to Rydberg blockade is described as

which means the phase shift is proportional to the Rydberg induced dispersion change Δn_r and the propagation distance L (or equivalently atomic density). See Methods for the theoretical derivations of Δn_r and ΔΦ₁(U).

For the two FWM signals (via the pathways and ) with the dressing effects of E₂ and E₄, the corresponding third-order polarizations P⁽³⁾ for the output FWM signals under steady-state condition are given by

,

,where N(v) = N₀exp(−v²/u²)/uπ^1/2 is the particle number density in terms of speed distribution function¹⁵; Ω_i = d_ijE_ij/ħ (i,j = 1, 2…) is the Rabi frequency between |i〉´|j〉 and d_ij is the dipole momentum; N₀ is the atom density; γ₁ = (Γ₁₀ + Γ_t) + i(Δ₁ + k₁v), γ₂ = (Γ₂₀ + Γ_c + Γ_t) + i(Δ₁ + Δ₂) + i(k₁ − k₂)v, γ₃ = (Γ₃₀ + Γ_t) + i(Δ₁ + Δ₃) + i(k₁-k₃)v, γ₄ = (Γ₄₀ + Γ_t) + i(Δ₁ + Δ₄) + i(k₁-k₄)v; Γ_ij = (Γ_i + Γ_j)/2 is the decoherence rate between |i〉 and |j〉; Γ_i is the transverse relaxation rate determined by the longitudinal relaxation time and the reversible transverse relaxation time; Δ_i = ω_ij − ω_i is the detuning between the resonant transition frequency ω_ij and the laser frequency ω_i of E_i. Note that the collision ionization rate Γ_c²⁵, transit time Γ_t and the Doppler effect (kv) should be considered. For the two EIT-assisted SWM signals via and , the corresponding fifth-order polarizations P⁽⁵⁾ are given by

,

.

Here, the additional phase factors e^iΔΦ and e^iΔΦ′ are introduced into the dressing terms (|Ω₂|/n¹¹)^0.4/γ₂ and |Ω₄|²/γ₄ to account for the propagation effect. ΔΦ = ΔΦ₁ + ΔΦ₂, where ΔΦ₁(U) is the phase modulation induced by the possibly coherent Rydberg-Rydberg interaction U; the relative phase ΔΦ₂ and ΔΦ′ are related to the orientations of induced dipole moments and can be manipulated¹⁸ 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 E₂. Suppressed and enhanced FWM1 signals (the suppressed condition is Δ₁ + Δ₄ = 0 and the enhanced condition is are observed by changing the frequency detuning of E₁ or E₄. According to the new two-photon dressed rule²⁶, 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 Ω₁ at Δ₁ = −Δ₄ = 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 E₁; and then |+₁〉 is split into |+₁±₂〉 secondly by E₂. Therefore, once the dressing level |+₁〉 moved around the position of Δ₁, 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 E₄ at Δ₁ = Δ₄ = 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 E₄ increases and the two-step dressing process can be simplified as level |1〉 is split into |±₁〉.

Figure 2

Dressed FWM1 process by scanning the frequency of Rydberg state (37D) coupling field E₂. (a) Switching between enhanced peak to suppressed dip by increasing Ω₁ at Δ₁ = −Δ₄ = 30 MHz. (b) Dependence of suppressed dip on Ω₄ at Δ₁ = Δ₄ = 0. (c) The probe field images versus Δ₁. (d1- d2) The E₄ EIT images without/with E₂ EIT dressing versus Δ₁ at discrete points of Δ₂ = Δ₄ = −Δ₁. (e1-e2) The dressed E₂ EIT and FWM1 images versus Δ₂ with Δ₁ = Δ₄ = 0. (f) Dressed energy level configurations with E₂ and E₁ dressing. (g-h) are the evolutions of dressed FWM1 versus Δ₂ by tuning Δ₁ at Δ₄ = 0 and tuning Δ₄ at Δ₁ = 0, respectively. (g1) and (h1) are corresponding theoretical predictions for (g) and (h) The Lorentzian profiles are the FWM1 signals versus Δ₁ and versus Δ₄. Ω₁ = 2π × 54 MHz at 0.5 mW, Ω₂ = 2π × 7.6 MHz at 200 mW, Ω₄ = 2π × 142 MHz at 6 mW, Ω₄′ = 2π × 224 MHz at 15 mW. The atom density is 1.0 × 10¹² cm⁻³.

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 E₁ and E₂ turned on, Fig. 2(c) shows the focusing/defocusing effects of probe signal versus Δ₁. Nonlinear refractive index n_r is negative in the self-focusing medium (Δ₁ < 0) while positive in the self-defocusing one (Δ₁ > 0). Figure 2(d1,d2) show the probe images with E₁&E₄ and E₁&E₂&E₄ on versus Δ₁, respectively. With E₂ blocked, the focusing/defocusing effects of probe images at different Δ₁ + Δ₄ = 0 can be stronger than the effects in Fig. 2(c) due to the growing of absolute value of refractive index. With E₁&E₂&E₄ on, the images of dressed E₄ EIT become more defocusing compared with the corresponding ones in Fig. 2(d1) due to Δn_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(d2) can be attributed to , where ΔΦ₁ can be modified as

Figure 2(e1,e2) are the images of dressed E₂ EIT and Rydberg dressed FWM1 versus Δ₂, respectively. The dressed FWM1 and dressed E₂ EIT with Δ₁ = Δ₂ = Δ₄ = 0 are much more defocusing than the points of Δ₂≠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 Δ₁ at Δ₄ = 0. The Lorentzian profile (curve constituted of the baseline of each signal) is a one-photon peak of the FWM1 signal versus Δ₁ and can be described by the single-photon term γ₁ in Eq. (3). The intensity of FWM1 in Fig. 2(g) is first suppressed and then enhanced at Δ₁ = −32 MHz, while it is first enhanced and then suppressed at Δ₁ = 32 MHz. Obviously, a dressing asymmetry occurs with Δ₁ = 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 Δ₁ > 0 and Δ₁ < 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; α₁ and α₂ are the ratio parameters for phase shift ΔΦ and ΔΦ′caused by E₂ and E₄, respectively.

According to Eq. (8), the value of A_F in Fig. 2(g) is about 0.58 at |Δ₁| = 32 MHz. Due to the absence of Autler-Townes (AT) splitting on the profile, the dressing effect of E₄ on the one-photon term γ₁ 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 Δ₁, one can attribute the results to the dressing effect of E₂ on γ₁. 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 [γ₁ + (|Ω₂|²/n¹¹)^0.4e^iΔΦ/γ₂]²γ₄ and can explain Fig. 2(g) well by setting ΔΦ = ΔΦ₁(U) + ΔΦ₂ = −π/3. (see Fig. 2(g1)).

Figure 2(h) is the modulated enhancement and suppression of FWM1 signal by increasing Δ₄ at Δ₁ = 0 and A_F is about 0.91 at |Δ₄| = 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 Δ₄, which can be described by the two-photon term γ₄ in Eq. (3). Obviously, the change of Δ₄ can also affect the modulated results of FWM1 and it can be ascribed to the dressing effect of E₂ on γ₄ associating with self-dressing shown in Eq. (3). As a consequence, the denominator of Eq. (3) is simplified as , which can account for Fig. 2(h) with ΔΦ = ΔΦ₁(U) + ΔΦ₂ = −π/3 and ΔΦ′ = −π (see Fig. 2(h1)).

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 Δ₄ at discrete Δ₁. To be specific, Fig. 3(a) is the case with E₂ beam blocked and shows the dressing effect of E₄ on FWM2 versus Δ₄ at different Δ₁, which can be well simulated by Eq. (4) by setting ΔΦ′ = −π/6 at Δ₃ = 150 MHz (see Fig. 3(a1)). As defined above, the dressing asymmetry factor A_F in Fig. 3(a) is 0.19 at |Δ₁| = 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 Δ₁ (see the one-photon term γ₁ in Eq. (4)) with E₂ blocked and the peak is broadened to be 200 MHz by the Doppler effect Δ₁−Δ₃ = k₁v + k₃v. Figures 3(b,c) are the ones with the dressing effect of E₂ (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 Δ₁. 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 |Δ₁| = 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 E₂ dressing effect and the cooperative atom-light interaction²⁷. 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 N₀ = 1 × 10¹² cm⁻³ and N₀ = 2.4 × 10¹² 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.

Figure 3

The change in Δ₁ induced enhancement and suppression of FWM2 together with the SWM2 process by scanning Δ₄ (a) without E₂ and (b) with E₂ coupling the transition between 5P_3/2↔54D_5/2, respectively, at atom density N₀ = 1 × 10¹² cm⁻³. (c) is the same to (b) except for N₀ = 2.4 × 10¹² cm⁻³. The profile (curve constituted of the baseline of each signal) in each panel is the FWM2 signal versus Δ₁, which is broadened by the Doppler effect Δ₁−Δ₃ = k₁v + k₃v. (a1)(b1) and (c1) are the theoretical predictions corresponding to (a)(b) and (c), respectively. (a1) ΔΦ′ = −π/6. (b1) ΔΦ′ = −π/6, ΔΦ = ΔΦ₁ + ΔΦ₂ = −π/12. (c1) ΔΦ′ = −π/6, ΔΦ = ΔΦ₁ + ΔΦ₂ = −π/3. Δ₂ = 0, Δ₃ = 150 MHz. Ω₁ = 2π × 54 MHz at 0.5 mW, Ω₂ = 2π × 7.6 MHz at 200 mW, Ω₄ = 2π × 116 MHz at 4 mW, Ω₃ = 2π × 170 MHz at 5 mW, Ω₃′ = 2π × 275 MHz at 13 mW.

Comparing the fourth curve in Fig. 3(b) with the fourth one at Δ₁ = 80 MHz in Fig. 3(a), the difference between the modulated results can be explained well by setting ΔΦ = ΔΦ₁ + ΔΦ₂ = −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 ΔΦ = ΔΦ₁ + ΔΦ₂  = −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 ΔΦ₁ caused by the change of cooperative nonlinearity. Considering that the values of ΔΦ₂ 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 Δ₁ at Δ₂ = Δ₃ = 0, respectively. The peaks of the FWM2 and SWM1 signals versus Δ₁ for 37D and 54D are shown by the Lorentzian profiles, which can be described by the one-photon term γ₁ in Eqs. (4) and (5), respectively. Since the results are related to the changing of Δ₁, they can be attributed to the dressing effects on the one-photon term γ₁ as shown in Eqs. (4) and (5). The phase shift of ΔΦ′ on the dressing term |Ω₄|²/γ₄ 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

Dressed MWM processes by scanning Δ₄. (a-b) The enhancement and suppression of FWM2 and SWM1 dressed by the SWM2 process with Δ₁ growing at Δ₂ = Δ₃ = 0 for 37D and 54D, respectively. The Lorentzian profiles (curve constituted of the baseline of each signal) are the FWM2 and SWM1 signal versus Δ₁ for 37D and 54D, respectively. (c-d) The enhanced and suppressed SWM1 by increasing the Δ₂ with Δ₁ = −Δ₃ = 30 MHz for 37D and 54D, respectively. The Lorentzian profiles are the SWM1 signal versus Δ₂ for 37D and 54D. (a1-d1) are the theoretical curves corresponding to (a-d) with ΔΦ′ = −π/6, respectively. The Rydberg-induced phase shift difference between 37D and 54D is about π/4. N₀ = 1 × 10¹² cm⁻³↔Ω₁ = 2π × 54 MHz at 0.5 mW, Ω₂ = 2π × 7.6 MHz at 200 mW, Ω₄ = 2π × 116 MHz at 4 mW, Ω₃ = 2π × 170 MHz at 5 mW, Ω₃′ = 2π × 275 MHz at 13 mW.

Figure 4(c,d) are the enhanced and suppressed SWM1 for 37D and 54D at Δ₁ = –Δ₃ = 30 by altering Δ₂, respectively. The Lorentzian profiles with linewidth of 60 MHz are the two-photon peaks of the SWM1 signal versus Δ₂ for 37D and 54D, respectively and related to the two-photon term γ₂ 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 γ₂ whereas the dressing effects on γ₁ 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 Δ₂ is not as obvious as in Y-type system by changing Δ₄. 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 Δ₁ = –120 MHz in Fig. 4(a,b) as the functions of the probe field strength P₁, the Rydberg state coupling field strength P₂ and the coupling field strength P₄ for three nD_5/2 states. We expand Eqs. (3)~(6) as Taylor series based on the dressing fields. Taking Eq. (5) as an example, we have

,where . Therefore, the intensities of the enhanced peak, suppressed dip and background are related to the trems , and , respectively. In addition, we have I∝n⁻³ according to Rydberg dressed MWM intensity I∝|Ω₂|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 state²⁷.

Figure 5(a) presents the E₁ 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 E₄ at Δ₁ + Δ₄ = 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 E₁. The saturating dressing-effect of E₄ means the phase shift is mainly caused by the existence of E₂ dressing and blockaded effect.

Figure 5

Power dependences (P₁, P₂ and P₄ 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. N₀ = 1 × 10¹² cm⁻³. Δ₁ = Δ₂ = 0, Δ₃ = 150 MHz. Ω₃/2π = 170 MHz at 5 mW, Ω₃′/2π = 275 MHz at 13 mW. (a1-a3) Ω₂/2π = 7.6 MHz at 200 mW, Ω₄/2π = 116 MHz at 4 mW. Ω₁/2π grows from 0 to 68 MHz at 0.8 mW. (b1-b3) Ω₁/2π = 54 MHz at 0.5 mW, Ω₄/2π = 116 MHz. Ω₂/2π grows from 0 to 7.6 MHz at 200 mW. (c1-c3) Ω₁/2π = 54 MHz, Ω₂/2π = 7.6 MHz, Ω₄/2π grows from 0 to 259 MHz at 20 mW.

The cases of the E₂ power dependences for three nD_5/2 states are shown in Fig. 5(b). First, we focus on the P₂ 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 Ω₂ increases, the enhanced SWM1 signal also makes the height of enhanced peak increase. However, the blockade term (|Ω₂|/n¹¹)^0.4 makes the curve saturated at higher power level. Then, the descending part of the curve is due to the dressing effect of E₂ associated with its excitation blockade effect from |Ω₂|/n¹¹)^0.4e^iΔΦ/γ₂. Next, the power dependence of the suppressed dip can also reflect the blockade effect and the dressing effect of E₂ (see Fig. 5(b2)). Initially, the saturated dressing of E₄ on FWM2 signal at Δ₁ + Δ₄ = 0 and gradually increased SWM1 signal are the main factors. Then the curve becomes saturated due to the blockade term (|Ω₂|/n¹¹)^0.4 at higher power level of E₂. 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 P₂ 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 E₄. The saturation is due to the blockade effect and the descending part is due to the combination of blockade effect and dressing effect of E₂. Given the above descriptions and analysis of peak and dip evolution corresponding to P₂ strength dependence, one can deduce that the phase modulation as well as asymmetry can become more obvious by strengthening E₂. 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, E₄ power dependences in Fig. 5(c) just show the regular enhancement and suppression processes by E₄. The asymmetry changes are mainly aroused from the dressing effect of E₄.

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

The first-order differential can describe spatial shift/splitting effects and the second-order differential 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)(α₁ΔΦ + α₂ΔΦ′) is established to depict the phase shift between dressing dark and bright states, where ΔΦ includes the phase shifts from both Rydberg dressing states and Rydberg excitation blockade and ΔΦ′ results from the orientations of induced dipole moments. The parameters α₁ and α₂ 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 D₂ line is driven by weak laser beam E₁ stabilized to a temperature-controlled Fabry-Perot (FP) cavity. A pair of coupling beams E₃ and E₃′, also driving the transition of D₂ line for different hyperfine configuration, are from another ECDL locked to the saturated absorption signal of rubidium atom. Beam E₂ driving the Rydberg excitation is a frequency-doubled laser with high stability. We get the needed 480 nm laser E₂ 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 E₂ adding onto the beam E₃ (in the same direction), which counter-propagates with beam E₁, drives the highly-excited Rydberg transition. E₄ and E₄′ are from the same LD4. E₄ adds onto the beam E₃ by a cubic polarizing beam splitter (PBS) and E₄′ propagates with E₃′ symmetrically with respect to E₂. 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 × 10¹² cm⁻³.

Theoretical models for Δn_r and ΔΦ₁(U)

Nonlinear refractive index change is modeled by taking Δn_r as the product of the slope of the dispersion (∂n_r/∂ω₂) and the energy level shift (Δω₂) of the Rydberg state due to the Rydberg-Rydberg interaction. ∂n_r/∂ω₂ is derived from the real part of the complex susceptibility²⁸ χ for stationary atoms and zero-coupling detuning as

where n₀ is the linear refractive index and D₁ = γ₁ + |Ω₂|/γ₂ + |Ω₄|/γ₄. The energy level shift is

where U(r − r′) is the cooperative nonlinear interaction for Rydberg atoms at nD states; N₂ is the density of excited Rydberg atoms. If we calculate the Rydberg excitation density via optical Bloch equation (OBE) by using the mean-field model² and taking N₂V_d = 1 & V_d  ∝(R_d)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, . So the density of excited Rydberg atoms N₂ is given as, where N₁ is the density of atoms at level |1〉. With the EIT effects and optical pumping effect taken into consideration, N₁ is given by , where γ₃₁ = Γ₁₃ + iΔ₃; 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

The induced phase modulation under the cooperative nonlinear interaction is