Eight-wave mixing process in a Rydberg-dressing atomic ensemble

We investigate the eight-wave mixing (EWM) process involving highly excited Rydberg states with the assistance of coexisting electromagnetically induced transparency (EIT) windows in a thermal Rb vapor both theoretically and experimentally. By use of a disturbance-free optical detection method, the Rydberg EWM characterized by multiple sets of spin coherence is presented via the interplay and competition between the dressing-state effects and excitation blockade caused by strong RydbergRydberg interaction. Such interplay and competition can be demonstrated by the intensity evolutions of multi-wave mixing (MWM) signals via controlling the atomic density, the frequency detuning and Rabi frequencies of corresponding laser fields. The observed Rydberg EWM tailored by EIT windows can possess of much narrower linewidth <30MHz and provide a new way for the study of Rydberg effect in the atomic ensemble above room temperature. ©2015 Optical Society of America OCIS codes: (020.5780) Rydberg states; (190.4223) Nonlinear wave mixing; (030.1640) Coherence. References and links 1. T. F. Gallagher, Rydberg Atoms (Cambridge University, 1994). 2. D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93(6), 063001 (2004). 3. K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemüller, “Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms,” Phys. Rev. Lett. 93(16), 163001 (2004). 4. M. Saffman, T. G. Walker, and K. Molmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010). 5. E. Urban, T. A. Johnson, T. 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Introduction
With the valence electrons excited to high-lying states, the Rydberg atoms possess of many exaggerated properties such as huge polarizability that scales as n 7 (n is the principle quantum number), long-range interaction and long lifetime [1].Such properties enable one Rydberg atom to block the excitations of other nearby atoms with strong and tunable van der Waals or dipole-dipole interaction [2,3].The promising applications of the blockade effect results from the Rydberg energy level shifting cover the range of quantum computation [4,5], atomic clocks [6], single-photon applications [7,8] and room-temperature quantum devices [8].The desirable way to obtain the reliable and controllable Rydberg effects is effectively driving the ground-state atoms to the Rydberg states.For example, electromagnetically induced transparency (EIT) window with much narrower linewidth [9][10][11][12][13][14] is used to probe Rydberg atoms with high resolution [15][16][17][18][19][20][21] and multi-wave mixing (MWM) process is applied for nondestructively optical detection [22,23].By combining the above EIT and MWM techniques together, we can observe the EIT-assisted Rydberg multi-waving mixing (MWM) process whose coherence time is shorter than ionization time [24].Actually, the incoherence plasma formation in a Rydberg gas has been reported to be on a time scale of ~100 ns or larger [25,26].In addition, the spatial arrangement of such EIT configuration can significantly suppress the Doppler effect, which makes the hot atoms in the beam volume behave like cold atoms to some extent [9].
In this paper, we theoretically and experimentally study the eight-wave mixing (EWM) process and fluorescence signal corresponding to the EIT window involving in Rydberg states in a thermal 85 Rb vapor.With the blockaded and non-blockaded EIT windows generated effectively, we can observe the Rydberg EWM process together with other MWM processes and corresponding high-order fluorescence signals.Particularly, a disturbance-free and controllable blockaded EWM process can be picked up by scanning the frequency of the Rydberg field.The EWM process in the thermal Rydberg-EIT medium can be manipulated via the atomic density, frequency detuning and Rabi frequencies of the laser fields.The coexisting EIT windows can tailor the MWM signals that carry the blockade information with a narrower linewidth [27], which can provide a new method for the study of Rydberg effect in the atomic vapor above room temperature.In the meanwhile, Rydberg blockade property also creates saturation for such EWM process, which can improve the controllability of multichannel quantum effect related to Rydberg states by controlling the spin coherence [28].We use four stabilized external cavity diode laser (ECDL) systems with stable continuouswave output to generate the Rydberg EWM process in a five-level 85 Rb atomic system, which consists of two hyperfine states F = 3 (|0) and F = 2 (|3) of ground states 5S 1/2 , a first excited state 5P 3/2 (|1) and two second excited states 5D 5/2 (|4) and nD 5/2 (|2, n = 37, 45).The highly-excited Rydberg states nD are directly related to the blockaded EWM process and fluorescence signal.Five beams derived from the four laser systems with linewidthes <1MHz at 5us are coupled into a 10 mm long rubidium cell wrapped with μ-metal sheets and heated by the heating tape to connect the corresponding transitions shown in Fig. 1(a).A weak probe laser E 1 (wavelength of 780.2 nm, frequency ω 1 , wavevector k 1 ) stabilized to a temperaturecontrolled Fabry-Perot (F-P) cavity drives the transition |0↔|1, while E 3 (780.2nm, ω 3 , k 3 ) and E 3 ′ (ω 3 , k 3 ′) from the same LD locked to the D2 line of rubidium connect |3↔|1.The strong beam E 2 (~480 nm, ω 2 , k 2 ) produced by frequency doubling a 960nm ECDL counterpropagates with beam E 1 to excite the Rydberg transition |1↔|2.Beam E 4 (775.9nm, ω 4 , k 4 ) driving transition |1↔|4 propagates with E 3 ′ symmetrically with respect to probe beam.By turning light beams on or off selectively, one can obtain different MWM process and highlight the blockaded EWM process.The probe field, MWM spectrum and fluorescence signals are received by D1, D2 and D3, respectively.

Basic theory
The mean-field model in Fig. 1(c) is employed to interpret the level shift ε of Rydberg states caused by the strong Rydberg-Rydberg interaction [2,3,29].To be specific, the region where has only a single Rydberg atom can be viewed as a sphere with a radius of R d , which determines the Rydberg atomic density in the beam volume.With atomic density N 2 at |2 considered to be locally uniform inside and around the given sphere with volume By solving the nonlinear Bloch-like equations for the three-level system (|0↔|1↔|2), we obtain the shift ε associated with the principal quantum number n and the location r in the sphere as where U(r−r′) is the Rydberg-Rydberg interaction for 85 Rb atoms at nD states.With the optical Bloch equations (OBEs) for the Rydberg excitation (5S 1/2 ↔5P 3/2 ↔nD 5/2 ) solved under steady-state condition and where C is a constant resulting from numerical integration outside the given sphere and the atom excitation efficiency between |0↔|1 and mainly determined by the coefficient of van der Waals interactions with its dependence on n eliminated, N 1 is the atomic density at |1 and G 2 is the Rabi frequency of E 2 .Considering of the coexisting EIT effects of E 2 and E 4 in |0↔|1↔|4 and |0↔|1↔|4 and optical pumping effect of E 3 in |0↔|1↔|3, N 1 can be described as ( ) where ) is the Rabi frequency for transition |i↔|j with μ ij being the dipole momentum;

⎯⎯⎯ →
To be specific, we take the threelevel subsystems |0↔|1↔|2 and |0↔|1↔|4 for instance to compare the atomic densities in Rydberg level |2 and non-Rydberg level |4.Because the two driving fields can be viewed as dressing fields for the first-excited level |1, the atomic density expressions for the two levels can have the similar formats and we can obtain the previous correspondences.
The intensities of the EIT signal and fluorescence signals in the atomic system are proportionally to the imaginary part of the first-order density matrix element ρ 10 (1) and the real matrix elements ρ 11 (2) and ρ 44 (4) , respectively.Via perturbation chain ( ) ( ) with the dressing effects and blockade effect is given by Via perturbation chain (2) 00 10 11 , the dressed fluorescence signal (2)   11 ρ can be described as By , we can obtain the blockade EWM signal corresponding to the seventh-order polarization P (7) .Considering of the dressing effects of E 1 , E 2 & E 4 and the Doppler effect (kv), P (7) is given by 0 where is the atomic density in terms of velocity distribution, Γ c is the collision ionization rate and Γ t is the transit time [30].The optical fields terms N 0 0.2 |G 1 | 0.4 and (|G 2 |/n 11 ) 0.4 in Eq. ( 6) describe the suppressions of the contributions from the population of state |1 and the excitation capacity of E 2 , respectively, to the Rydberg excitation, due to the blockade effect.With the Doppler effect (kv) taken into consideration, the one-photon term should be Then with the introducing of the collision ionization rate Γ c ( = Aν rel σ g ρ 2 , where A is a constant, ρ 2 = N 2 /πlw 0 2 and N 2 is the number of Rydberg atoms, l and w 0 are the length and radius of interaction region, v rel is the relative velocity, and σ g = π(a 0 n 2 ) 2 ), the Rydberg two-photon term is modified as

The verification of the Rydberg EWM process
The Rydberg EWM process corresponding to three transitions (|0↔|1, |2↔|1 and |4↔|1 with linear susceptibilities (1)   10 χ , (1)   21 χ and (1)   41 χ , respectively) can be verified by the interaction and dressing effects between the E 2 and E 4 dressing fields, which are demonstrated via the intensity evolutions of the EWM channel and corresponding EIT and fluorescence channels with the frequency detuning of E 2 scanned.By scanning Δ 2 at different Δ 4 , the EIT, fluorescence and EWM signals with high signal-to-noise ratio (SNR) are shown in Fig. 2. For the EIT peak (satisfying Δ 1 + Δ 2 + ε = 0) created by E 2 (n = 37) in Fig. 2(a), the height and the full width at half maximum (FWHM) of each peak represent the intensity and linewidth, respectively.When Δ 4 (with the range from −200 MHz to 200 MHz) moves towards the point zero, the E 2 EIT signal becomes weaker due to the increase of dressing term |G 4 | 2 /d 4 in Eq. ( 4), where d 4 = Γ 40 + i(Δ 1 + Δ 4 ) and Δ 1 are fixed.One can deduce that the dressing effect of E 4 becomes stronger so that the suppression effect on E 2 EIT can be more obvious.Figure 2(b) shows the fluorescence signals corresponding to the EIT peaks in Fig. 2(a).The secondorder fluorescence emitting from |1 are modulated by both E 2 and E 4 , which can result in the interaction and competition between the two dress modulations.For the case of E 4 dressing, there is also a high-order fluorescence peak mainly emits from |4 located inside the dip while there only exists a dip for the case of E 2 .The difference mainly results from the long lifetime of the Rydberg state, which can render the Rydberg fluorescence peak too weak to be observed.According to Fig. 2(b), it is obvious that the depth of the dip that signifies the intensity of the Rydberg fluorescence becomes smaller with Δ 4 closer to the resonant point (satisfying Δ 1 + Δ 2 = 0 and Δ 1 + Δ 4 = 0), which can be attributed to Eq. ( 5).When   2(c), the EWM signal versus Δ 2 becomes stronger with Δ 4 approaching to zero.And the obvious increment of P (7) comes from the dressing enhancement on E 2 caused by E 4 according to Eq. ( 6).We add two dashed lines in Fig. 2(c), the upper one is consisted of the peak of Rydberg SWM1&EWM signals and the lower one is the profile of a single non-Rydberg SWM2 peak versus the probe detuning Δ 1 .With Δ 4 being far from resonance, the intensity of SWM2 is near to zero and the observation can be pure Rydberg SWM1 (k . At the resonance points, the peaks become obviously higher than those at large frequency detuning points.The intensity evolution can also be explained by Eq. ( 6).When the E 4 EIT window overlaps with the Rydberg EIT window at Δ 4 = 0, the two-photon resonance conditions Δ 1 + Δ 2 = 0 and Δ 1 + Δ 4 = 0 are satisfied.Due to the terms d 4 ′ = (Γ 40 + Γ t ) + i(Δ 1 + Δ 4 ) + i(k 1 −k 4 ) and d 4 = Γ 40 + i(Δ 1 + Δ 4 ) in the denominator of Eq. ( 6) can be the minimum value, the enhancement interaction of E 4 on E 2 can be the strongest and the seventh-order polarization can be the maximum.Such increment fraction between the two dashed lines can turn out the existence of Rydberg EWM process effectively.At low E 4 power, only the SWM1 signal can be obtained and the intensity is small due to the weak E 4 dressing enhancement on E 2 .Then, the signal intensity can be strengthened apparently with E 4 power increasing from 0 to 25mW, which leads to the increase of G 4 in Eq. ( 6).However, the Rydberg MWM intensity begins to decrease with E 4 power over 25mW, which means the suppression is getting stronger than the enhancement.Comparing the amplitudes at P 4 = 0 and P 4 = 25mW in Fig. 3(a1), we can identity the intensity of Rydberg EWM.As a result, one can find that the intensity of Rydberg EWM can be quantitatively controlled by G 4 .

Controlling the Rydberg EWM process
Figures 3(b) and 3(c) depict the interactions as well as dressing effects between the two overlapped EIT windows and corresponding fluorescence signals, respectively.As observations (squares) shown in Fig. 3(b), the strength of EIT signal can become smaller when E 4 power alters from 0 to 30mW, which agrees well with the theoretical curve.The theoretical prediction illustrates that the suppression on E 2 EIT spectra becomes stronger due to the rising G 4 .Figure 3(c) shows the measured Rydberg fluorescence dip can become smaller with P 4 increasing, which coincides well with the simulation.

The blockade effect in the Rydberg EWM process
Finally, we focus on the illustration of the excitation blockade effect in the Rydberg EWM process.According to seventh-order polarization intensity shown in Eq. ( 6), we have Rydberg EWM intensity I∝n −6 with I∝|μ ij | 4 and μ ij ∝n (−3/2) taken into account.Consequently, the intensities of the Rydberg signals in Fig. 4 are scaled to n = 37 by the factor (n*/37*) 6 to eliminate the dipole transition μ ij influence for different n.Here, n* = n−δ, and δ = 1.35 is due to the quantum defect for nD 5/2 state [31][32][33].At Δ 1 = Δ 4 = 0, Figs.4(a) and 4(b) show that the intensities of the EIT and fluorescence signals can be modulated by adjusting E 2 power P 2 for 37D (squares) and 45D (dots), respectively.As G 2 increasing, the blockade effects on the EIT and fluorescence spectra appear gradually and the intensity difference between n = 37D and 45D with same P 2 also becomes larger.We can also find that the blockade effect becomes stronger with larger n.The EIT and fluorescence intensity evolutions can be understood with  Different from the cases of EIT and fluorescence signals, the P 2 dependence of EWM for 37D (squares) and 45D (dots) in Fig. 4(c) gets saturated and even declined due to the combined dressing effects of E 1 and E 4 described in Eq. ( 6).It means that the evolution of the EWM process is more sensitive to the dressing and blockade effects since in numerator term can have a more obvious influence than that in the denominator term in Eq. (6).Meanwhile, the atomic density dependences of the EWM for 37D (squares) and 45D (dots) obtained by varying the atom temperature are exhibited in Fig. 4(d), which agrees well with the corresponding theoretical predictions.We can find that there exists an obvious intensity saturation even attenuation after the largest intensity point.Such a saturated even declined trend is completely derived from the blockade term N 2 proportional to N 0 0.2 , which can lead to the saturation when N 0 increases.The pure EWM in Fig. 4(e3) can be picked up when the coexisted Rydberg EWM&SWM1 in Fig. 4(e1) minus the pure SWM1 in Fig. 4(e2).With the non-Rydberg driving field E 4 turned off, there only exists the Rydberg driving field as well as the pure SWM1 signal shown in Fig. 4(e2).Due to the scissor effect caused by the overlapping EIT windows, the Rydberg EWM process possesses an ultra-narrow linewidth less than the respective linewidths of the related EIT signals as well as the two SWM signals.Also, the ultra-narrow EWM signal is much more sensitive to the excitation blockade and the multi-dressing properties.The linewidth of the EWM is described as Δω E = Δω 10 (Δω 21 /Δϖ 21 )(Δω 41 /Δϖ 41 ), where Δω ij is the linewidth of EIT window suppressing the absorption of the field exciting the transition |i↔|j, while Δϖ ij is the linewidth of absorption spectrum without dressing effect.In contrast to the Rydberg EIT, the Rydberg EWM spectrum provides higher resolution due to its narrower linewidth, which indicates that the EWM is more sensitive to the Rydberg blockade.

Conclusion
With the Rydberg and non-Rydberg EIT windows generated effectively in thermal vapor environment, we theoretically and experimentally investigate the multi-EIT-assisted Rydberg EWM process with excitation blockade.The blockade information caused by the strong Rydberg-Rydberg interactions is mapped into such EWM process, which can provide a reliable blockade control in the multi-level atomic system.The typical features of the Rydberg EWM process with high SNR are demonstrated by characterizing it as a function of coupling field strength and density of collective atom ensemble by scanning the frequency detuning of dressing fields.Such blockaded EWM process with ultra-narrow linewidth and sensitive response to the blockade may provide a new way to study the Rydberg effect in the atomic vapor above room temperature.

Fig. 1 .
Fig. 1.(a) Experimental setup.L-lens, D-photodetectors, SAS-saturated absorption spectrum, TA-tapered amplifier, FP-Fabry Perot cavity, FD-frequency doubler, DL-external cavity diode laser, HW-half wave plate at corresponding wavelength, PBS-polarized beam splitter at corresponding wavelength, OI-optical isolator.Double-headed arrows and filled dots denote horizontal polarization and vertical polarization of incident beams, respectively.(b) A fivelevel atomic system in 85 Rb vapor.(c) Model for realizing Rydberg excitation blockade.|2 is the Rydberg state level; the ellipse background represents a region within which only one atom can be excited to the Rydberg state; the top big ball represents the single Rydberg atom, while the small balls represent atoms in lower levels; ε 2 −ε is the shifted level energy.
Δ 4 is near to zero, the dressing effect on fluorescence dip by term |G 4 | 2 /d 4 gets larger, which indicates the stronger suppression on E 2 fluorescence.The background Lorentzian profiles (curve constituted of the baseline of each signal) in Figs.2(a) and 2(b) show the EIT and corresponding fluorescence signals created by E 4 with probe detuning Δ 1 scanned, which can be described by term |G 4 | 2 /d 4 in Eqs.(4) and (5).

Fig. 2 .
Fig. 2. (a)~(c) Intensity evolutions of EIT, fluorescence and corresponding Rydberg EWM signals by scanning Δ 2 with increased Δ 4 .The single peak in (a), dip in (b) and peak in (c) are the intensities of the EIT, fluorescence and MWM signals, respectively.The dashed lines in Fig. 2 (a) and Fig. 2(b) are the EIT and fluorescence profiles of E 4 field with the probe detuning Δ 1 scanned.The upper dashed line profile is consisted of the peaks of Rydberg SWM1&EWM signals.The lower dashed line is the profile of a single non-Rydberg SWM2 peak with the probe detuning Δ 1 scanned.(a1)~(c1) are the theoretical curves corresponding to (a)~(c), respectively.Level |2 is 37D.N 0 = 2.4 × 10 12 cm −3 , G 1 = 2π × 54 MHz, G 2 = 2π × 7.6 MHz, G 3 = 2π × 350 MHz, G′ 3 = 2π × 150 MHz, G 4 = 2π × 600 MHz.With the assistance of EIT and fluorescence signals, we obtain the pure Rydberg EWM process from the MWM channel by scanning the frequency detuning of Rydberg dressing fields.With Δ 4 increasing from −200 MHz to 200 MHz in Fig.2(c), the EWM signal versus Δ 2 becomes stronger with Δ 4 approaching to zero.And the obvious increment of P(7) comes from the dressing enhancement on E 2 caused by E 4 according to Eq. (6).We add two dashed lines in Fig.2(c), the upper one is consisted of the peak of Rydberg SWM1&EWM signals and the lower one is the profile of a single non-Rydberg SWM2 peak versus the probe detuning Δ 1 .With Δ 4 being far from resonance, the intensity of SWM2 is near to zero and the observation can be pure Rydberg SWM1 (k SWM1 = k 1 + k 3 −k 3 ′ + k 2 −k 2 ).At the resonance points, the peaks become obviously higher than those at large frequency detuning points.The

Figure 3
Figure 3 mainly demonstrates the influences of the power, namely, Rabi frequency, of dressing field E 4 on the EWM process experimentally (squares) and theoretically (solid curves).Firstly, Fig. 3(a) presents the E 4 power P 4 dependence of the EWM signal intensity by scanning Δ 2 .Figures 3(a1) and 3(a2) represent the experimentally measured and theoretically simulated lineshapes of corresponding signals, respectively, so do those shown in Figs.3(b1-b2) and 3(c1-c2).At low E 4 power, only the SWM1 signal can be obtained and the intensity is small due to the weak E 4 dressing enhancement on E 2 .Then, the signal intensity can be strengthened apparently with E 4 power increasing from 0 to 25mW, which leads to the increase of G 4 in Eq. (6).However, the Rydberg MWM intensity begins to decrease with E 4 power over 25mW, which means the suppression is getting stronger than the enhancement.Comparing the amplitudes at P 4 = 0 and P 4 = 25mW in Fig.3(a1), we can identity the intensity of Rydberg EWM.As a result, one can find that the intensity of Rydberg EWM can be quantitatively controlled by G 4 .Figures3(b) and 3(c) depict the interactions as well as dressing effects between the two overlapped EIT windows and corresponding fluorescence signals, respectively.As observations (squares) shown in Fig.3(b), the strength of EIT signal can become smaller when E 4 power alters from 0 to 30mW, which agrees well with the theoretical curve.The theoretical prediction illustrates that the suppression on E 2 EIT spectra becomes stronger due to the rising G 4 .Figure3(c) shows the measured Rydberg fluorescence dip can become smaller with P 4 increasing, which coincides well with the simulation.
, N 0 is the atomic density at ground state |0.By comparing Eq. (2) with that of the non- is the decoherence rate between |i and |j; Γ i is the transverse relaxation rate; Δ i = Ω i −ω i is the detuning between the resonant transition frequency Ω i and the laser frequencyω i of E i Blockade G G n