Study of Rydberg blockade mediated optical non-linearity in thermal vapor using optical heterodyne detection technique
SStudy of Rydberg blockade mediated optical non-linearity in thermal vapor usingoptical heterodyne detection technique
Arup Bhowmick, ∗ Dushmanta Kara, and Ashok K. Mohapatra School of Physical Sciences, National Institute of Science Education & Research, Bhubaneswar 751005, INDIA. (Dated: October 4, 2018)We demonstrate the phenomenon of blockade in two-photon excitations to the Rydberg state inthermal vapor. A technique based on optical heterodyne is used to measure the dispersion of a probebeam far off resonant to the D2 line of rubidium in the presence of a strong laser beam that couplesto the Rydberg state via two-photon resonance. Density dependent suppression of the dispersionpeak is observed while coupling to the Rydberg state with principal quantum number, n = 60. Theexperimental observation is explained using the phenomenon of Rydberg blockade. The blockaderadius is measured to be about 2 . µ m which is consistent with the scaling due to the Doppler widthof 2-photon resonance in thermal vapor. Our result promises the realization of single photon sourceand strong single photon non-linearity based on Rydberg blockade in thermal vapor. I. INTRODUCTION
Rydberg atoms are enriched with enhanced two-bodyinteractions. When atoms in a dense frozen ensemble areexcited to the Rydberg state using narrow band lasers,strong Rydberg-Rydberg interactions lead to excitationblockade. This blockade interaction generates a highlyentangled many-body quantum state in an ensemble ofatoms which has become the basis for fundamental quan-tum gates using atoms [1–4] or photons [5] and for real-ization of single photon source [6, 7]. Rydberg block-ade in ultra-cold atoms and Bose-Einstein condensate(BEC) has been proposed to study strongly correlatedsystems [8–11]. In addition, strong photon-photon inter-actions enabled by Rydberg blockade mediated opticalnon-linearity has been proposed [12, 13]. Rydberg block-ade has been demonstrated for an ensemble of ultra-coldatoms in a magneto-optical trap (MOT) [14–17] or in amagnetic trap [18, 19] and also for single atoms trappedin optical micro traps [20, 21]. Recently, strong opticalnon-linearity mediated by Rydberg blockade in cold atomhas been observed for weak classical light [22] as well asfor single photons [23, 24] and also in a cold atomic sam-ple inside an optical cavity [25].The blockade radius for an ultra-cold atomic sampleis defined as r b = (cid:113) C (cid:126) Ω eff , where Ω eff is the effectiveRabi frequency of the Rydberg excitation and C is thestrength of the van der Waals type Rydberg-Rydberg in-teraction. For a thermal ensemble of atoms, the block-ade radius is affected by the Doppler width (∆ ν D ) dueto the thermal motion of atoms. Since the van der ∗ E-mail: [email protected] Lo w pa ss f il t e r W a v e f o r m M i x e r ω - δ λ /4 platePBSAOM ω OscilloscopeOscilloscope780 nm480 nm P o l a r i z e r λ / p l a t e P o l a r i z e r AOM ω - δω Δ p |g ⟩ |e ⟩ |r ⟩
480 nm780 nm σ + σ - σ + δ (a) (b) Δ c -3 -2 -1 0 1 20.00.51.0 ∆ φ ( a . u . ) ∆ C (GHz) ,F=3 → nS ,F=2 → nS (c) FIG. 1. (a) Schematic of the experimental set up. (b) Energylevel diagram for 2-photon transition to the Rydberg state.Two probe beams with σ + and σ − polarizations couple thetransition 5s / , F= 3 ( | g (cid:105) ) −→ / ( | e (cid:105) ) of Rb. The cou-pling laser with σ + polarization couples the transition 5p / ( | e (cid:105) ) −→ n s / ( | r (cid:105) ). The probe(coupling) detuning is ∆ p (∆ c )and frequency offset between the probe beams is δ . (c) Typ-ical dispersion spectrum of a probe beam by scanning thecoupling over 5 GHz. Waals interaction scales as the sixth power of the inter-atomic separation, the blockade radius is scaled down as r b ∝ √ ∆ ν D [26]. For thermal rubidium atoms at roomtemperature, the blockade radius is decreased only by afactor in the range of 2 −
3. If one works with a Rydbergstate, n >
60, the blockade radius in thermal vapor is ofthe order of a few microns. Electromagnetically inducedtransparency involving Rydberg state (Rydberg EIT) hasbeen demonstrated in thermal vapor cell [27] and in mi-cron sized vapor cell [26]. Van der Waals interactions of a r X i v : . [ phy s i c s . a t o m - ph ] M a y Rydberg atoms in thermal vapor has been observed re-cently [28]. In addition, four wave mixing involving Ryd-berg state [29] and large dc Kerr non-linearity of a Ryd-berg EIT medium has also been demonstrated in thermalvapor [30].In this article, we present the first ever strong evidenceof Rydberg blockade in thermal vapor using narrow-bandlasers for Rydberg excitations. The schematic of the ex-perimental set up is shown in figure 1(a). A techniquebased on optical heterodyne is used to measure the dis-persion of a probe beam due to two-photon excitationto the Rydberg state. A similar method used for dis-persion measurement in atomic vapors is reported in ref-erences [31, 32]. An external cavity diode laser operat-ing at 780 nm is used to derive two probe beams. Afrequency offset is introduced between the probe beamsby passing them through two acousto-optic modulators.Both the beams are made to superpose using a polar-izing cube beam splitter (PBS). The interference beatsignals are detected using two fast photo-detectors byintroducing polarizers at both the output ports of thePBS. The probe beams coming out of one of the out-put ports of the PBS propagate through a magneticallyshielded rubidium vapor cell. The density of Rb vaporcan be increased to 5 . × cm − by heating the cellup to 130 C. The coupling beam is derived from a fre-quency doubled diode laser operating at 478 −
482 nmand it counter-propagates the probe beams through thecell. The coupling and the probe beams are focused in-side the cell using lenses. The waist and the Rayleighrange of the probe (coupling) beams are 35 µ m (50 µ m)and 12 mm (10 mm), respectively. The peak Rabi fre-quencies of the laser beams and their variations over thelength of the vapor cell were calculated [33] using thesame parameters of the beams and were included in thetheoretical model to fit all the experimental data.The probe beams propagating through the medium canundergo different phase shifts by choosing suitable polar-izations of the probe and the coupling beams. The polar-ization of the coupling beam is chosen to be σ + and theprobe beams to be σ + and σ − as shown in figure 1(b).The probe beam with σ + polarization can not couplethe two-photon transition, 5s / → n s / and doesn’t gothrough any phase shift due to two-photon process. Weuse it as the reference probe beam. However, the otherprobe beam with σ − polarization can couple the sametwo-photon transition and hence, goes through a phaseshift due to two-photon excitation to the Rydberg state. - 2 . 0 - 1 . 5 - 1 . 0- 0 . 20 . 00 . 2 - 2 . 0 - 1 . 5 - 1 . 0024 Re( c ( a ) D C ( G H z ) ( b ) Re( c D C ( G H z ) FIG. 2. (a) Real part of the susceptibility of the mediumdue to 2-photon resonance. The peak Rabi frequencies of thecoupling beam was 24 MHz and of the probe beam was 60MHz (a) and 400 MHz (b). The lasers are coupled to the Ry-dberg state ( n = 30s / ). Open circles ( ◦ ) are the refractiveindex measured in the experiment, solid circles ( • ) are thecalculated refractive index using the exact model of a 3-levelatom interacting with a probe and a coupling beams. Solidlines are the calculated refractive index using an approximatemodel of considering the 3-level atom as an effective 2-levelatom. To compare with the theoretical model, the experi-mental data were scaled by a multiplication factor which canbe accounted for overall gain in the experiment. This additional phase shift of the signal probe beam ap-pears as a phase shift in the respective beat signal andis measured by comparing to the phase of the referencebeat signal detected at the other output port of the PBS.Since, both the beat signals are the output of the sameinterferometer, the noise due to vibration and acousticdisturbances are strongly suppressed. The signal-to-noiseratio was further improved by using a lock-in amplifier.The beat signals passing through a high pass filter havethe form, D α = A α cos( δt + φ α ), α = r, s . Here, δ isthe frequency offset between the probe beams and A r ( A s ), φ r ( φ s ) are the amplitude and phase of the refer-ence (signal) beat, respectively. A r and A s depend onthe power of the probe beams falling on the detector.The beat signals are then multiplied using an electronicwaveform mixer and are passed through a low pass fil-ter. The output of the low pass filter gives a DC signal, S L = 2 A r A s cos(∆ φ ) where ∆ φ = φ r − φ s . For anysmall variation of the phase around ∆ φ = 0 (∆ φ = π ), S L gives information about absorption (dispersion) of theprobe beam propagating through the medium. A phaseoffset between the beams falling on the reference detec-tor was introduced by placing a λ -plate before the polar-izer. As a result, the phase of the reference beat can becontrolled by rotating the polarizer axis. The refractiveindex of the σ − probe beam due to two-photon excita-tion to the Rydberg state can be measured with a phaseoffset, ∆ φ = π . The detailed principle of the techniquecan be found in reference [34]. For a larger frequency off-set between the probe beams in comparison to their Rabifrequency, the experimental result matches well with thestandard model of a 3-level atom interacting with a singleprobe and coupling beams [34]. Hence, a large frequencyoffset of 800 MHz is used in the experiment.In the experiment, the probe beam was stabilized at1 . Rb. The cou-pling laser frequency was scanned to observe the dis-persion of the probe beam by measuring its phase shiftdue to the two-photon excitations to the Rydberg state.A typical dispersion spectrum with the coupling beamscanning over 5 GHz is shown in figure 1(c). The 2-photon resonance peaks corresponding to the transitions,5s / F= 3 −→ n s / and 5s / F= 2 −→ n s / are ob-served and are used to normalize the frequency axis.The dispersion peak corresponding to the 5s / F= 3 −→ n s / transition was analyzed for the further study of Ry-dberg excitation. For a weak probe beam, an usual dis-persion profile of the two-photon resonance is observedas shown in figure 2(a). However, an absorptive like dis-persion profile is observed for a stronger probe beam asshown in figure 2(b). In order to explain the shape ofthe dispersion profile, we consider a three level atomicsystem interacting with two monochromatic laser field inladder configuration as shown in figure 1(b). Ω p and Ω c are used as probe and coupling Rabi frequencies respec-tively. The density matrix equation, i (cid:126) ˙ˆ ρ = [ ˆ H, ˆ ρ ]+ i (cid:126) L D ˆ ρ is solved numerically in steady state and is averagedover the Maxwell-Boltzmann velocity distribution of theatom. Here, ˆ H is the Hamiltonian for a three-level atominteracting with two mono-chromatic light field in a suit-able rotating frame and L D is the Lindblad operatorwhich takes care of the decoherence in the system. Inthe numerical calculation, we have used the decay rateof the channels | e (cid:105) → | g (cid:105) as 2 π × rg is used to account for the finite transit timeof the thermal atoms in the laser beams. The dipole ρ rg dephases at a rate of Γ rg + γ rel , where γ rel accounts forthe relative laser noise between the probe and the cou-pling beams. Γ rg and γ rel are of the order of 2 π × Peak height (a.u.) ( a ) W ( M H z ) ( b ) Normalized peak height W ( M H z ) FIG. 3. (a) Measured dispersion peak height as a functionof peak Rabi frequency of probe (Ω ) while coupling to theRydberg state ( n = 30s / ) with atomic vapor densities 2 . × cm − ( (cid:5) ), 1 . × cm − ( ◦ ) and 3 . × / cm − ( (cid:3) ).The peak Rabi frequency of the coupling beam was 13 . ate state [35]. Using the same approximation (∆ p >> Γ eg , Ω p and ρ ee ≈
0) in the steady state equations of3-level atom, the susceptibility of the medium can bewritten as χ = χ L + χ L , where χ L is the suscepti-bility of the lower transition without coupling beam and χ L is the susceptibility due to 2-photon resonance only. Re ( χ L ) = N | µ | (cid:15) (cid:126) (cid:16) − p (cid:17) where N is the vapor densityand µ is the dipole moment of the lower transition. Inthe regime Ω p >> Γ eg Γ rg , Re ( χ L ) = N | µ | (cid:15) (cid:126) (cid:18) p (cid:19) (cid:18) Ω p − p Ω p (cid:19) ρ rr (1)Where ∆ = ∆ + Ω p s − Ω c s with ∆ = ∆ p + ∆ c and∆ s = ∆ p − ∆ c . The Rydberg population ( ρ rr ) can bedetermined analytically using the effective 2-level model.Doppler averaging of the equation (1) with same laser pa-rameters fits well with the experimental data as shownin figure 2. This approximate model shows a very lit-tle deviation from the exact 3-level calculation for theprobe Rabi frequency up to 500 MHz. It is worthwhileto mention that Im ( χ L ) = N | µ | (cid:15) (cid:126) (cid:16) rg Ω p (cid:17) ρ rr . Compar-ing it with Re ( χ L ), the dispersion peak is an order ofmagnitude larger than the absorption peak for ∆ p ≈ p ≈
100 MHz.In a further study of variation of the dispersion peakheight as a function of probe Rabi frequency, an RF at-tenuator at the output of the detectors was used to keepthe amplitude of the beat constant irrespective of theprobe laser power. The variation of the dispersion peakheight as a function of probe Rabi frequency for different - 2 . 0 - 1 . 5 - 1 . 0- 0 . 0 80 . 0 00 . 0 80 . 1 6 - 2 . 0 - 1 . 5 - 1 . 0024- 2 . 0 - 1 . 5 - 1 . 00 . 00 . 81 . 6
Normalized peak height ( a ) ( b ) W ( M H z ) W ( M H z ) Re( c D C ( G H z ) ( c ) D C ( G H z ) ( e ) D C ( G H z ) ( d ) FIG. 4. (a) Normalized dispersion peak height as a function ofpeak Rabi frequency of the probe beam (Ω ) coupling to theRydberg state (60s / ) with atomic vapor densities 2 . × cm − ( (cid:5) ), 1 . × cm − ( (cid:3) ) and 3 . × / cm − ( ◦ ).The peak coupling Rabi frequency was 8 . . × / cm − . The dashedline is derived from the model without interaction. The datapoints are generated using the model which includes interac-tion induced dephasing only, but without blockade and withΓ rr = 100 MHz ( (cid:52) ), Γ rr = 500 MHz ( (cid:3) ), and Γ rr = 1000MHz ( (cid:5) ). The solid lines are derived from the model which in-cludes both interaction induced dephasing and blockade withΓ rr = 500 MHz and r b = 1 . µ m (light gray), r b = 2 . µ m (black), and r b = 2 . µ m (dark gray). The curve with r b = 2 . µ m matches well with the experimental data ( ◦ ).The dispersion spectra for the Rydberg state (60s / ) for thepeak probe Rabi frequencies equal to 50 MHz (c), 275 MHz(d) and 510 MHz (e). The green solid, red dashed, and blackdotted dashed lines are the dispersion spectra generated us-ing the model with both interaction induced dephasing andblockade, only with interaction induced dephasing but with-out blockade, and without interactions, respectively. vapor densities is shown in figure 3(a). The dispersionpeak height calculated by the theoretical model for thesame laser parameters and vapor densities agrees well asshown in figure 3(a). When the dispersion peak heightdata is normalized to that of a weak probe beam, thenall the data corresponding to different densities fall onthe same line as shown in figure 3(b). This observationsuggests that the refractive index of the medium dependslinearly on the vapor density and the Rydberg-Rydberginteraction has negligible effect.To study the blockade interaction, the blue laser wastuned to interact with a Rydberg state with principal quantum number n = 60 and the same experiment wasperformed. The variation of the dispersion peak heightat different densities in the strong Rydberg-Rydberg in-teraction regime is shown in figure 4(a). In contrastto the observation presented in figure 3, the normalizeddispersion peak height shows a non-linear dependenceof density and there is a clear indication of suppres-sion of the dispersion peak at higher densities. Since Re ( χ L ) ∝ ρ rr , then suppression in dispersion peakheight is due to the suppression of Rydberg populationwhich is the signature of the Rydberg blockade interac-tion. The ions inside the vapor cell has negligible effectwhich was confirmed using Rydberg EIT [27] for 60s / Rydberg state. The blackbody radiation induced ioniza-tion and transition rates are less than 10 kHz [36] andhas negligible effect. To exclude the possibility of sup-pression in the dispersion peak due to interaction induceddephasing, we introduced a Rydberg population depen-dent dephasing of the dipole matrix element ρ rg similarto the model discussed in reference [28]. Rydberg pop-ulation ( ρ rr ) decays at a rate of Γ rg and ρ rg decays ata rate of Γ rg + γ rel + ρ rr Γ rr . Introducing this term inthe effective 2-level model, a cubic equation of ρ rr is ob-tained. Looking at the coefficients of the cubic equation,one can deduce that out of the 3 solutions of ρ rr , oneis positive real and the other two solutions are eithernegative real or complex conjugate to each other. Thepositive solution of ρ rr is evaluated by solving the cubicequation numerically and is replaced in equation (1) tocalculate the dispersion of the probe beam. The disper-sion peak height calculated using this model is shown infigure 4(b). We have observed that depending on Γ rr ,the dispersion peak height reduces in comparison withthe non-interacting model, but monotonically increasesas the probe Rabi frequency and doesn’t display any fea-ture of saturation. Also, by increasing Γ rr from 500 MHzto 1 GHz, a very small reduction of the dispersion peakis observed.To explain the saturation of the experimental data, weintroduced blockade classically. Consider the number ofatoms per blockade sphere to be N b nad blockade radiusto be r b . The probability of simultaneous multiple exci-tations of n out of N b atoms to the Rydberg state is givenby P n = N b ! n !( N b − n )! ρ nrr (1 − ρ rr ) ( N b − n ) . For these events,only one Rydberg excitation out of n atoms is consideredand the blockaded Rydberg population is evaluated as ρ ( b ) rr = (cid:16) P + (cid:80) N b P n n (cid:17) ρ rr . The dispersion of the probeis then determined by replacing ρ ( b ) rr in equation (1). Thetheoretical curves generated after introducing blockadeinto the model are shown in figure 4. If we take r b = 2 . µ m, then it fits well with the experimental data for allthree densities. As shown in figure 4(b), if the blockaderadius is changed by 25%, it clearly doesn’t fit to theexperimental data. The width of the dispersion spectrafor different probe Rabi frequencies are shown in figure4(c,d and e). For Ω = 50 MHz, the Rydberg popu-lation is small and in this non-interacting regime, themodel with blockade and population dependent dephas-ing shows very little deviation from the non-interactingmodel. For Ω = 275 MHz, the model with populationdependent dephasing reduces significantly from the non-interacting model but further including blockade doesn’tshow much deviation. For Ω = 510 MHz, there is a sig-nificant suppression of the peak due to blockade. How-ever, the width of the dispersion spectrum doesn’t changeappreciably compared to the non-interacting model asshown in figure 4(e) which indicates a clear evidence ofRydberg blockade in our system.In conclusion, Rydberg blockade is demonstrated inthermal vapor and the blockade radius is measured to be 2 . µ m. The van der Waals coefficient ( C ) for the Ry-dberg state (60s / ) is 140 Ghz/ µ m [22]. The typicalwidth of the 2-photon resonance is about 500 MHz whichcan be determined from figure 1(c). Hence, blockade ra-dius in this system should be approximately 2 . µ m whichdiffer from our experimental result by less than 15%. It isto be noted that we have introduced blockade classicallyand a full quantum mechanical model may give a betterestimate for the blockade radius. Our result shows thatthe coherent collective Rydberg excitation is possible inthermal vapor and opens up the possibility to build thequantum devices like single photon source and photonicphase gate based on Rydberg blockade non-linearity inthermal vapor. II. ACKNOWLEDGMENTS
We acknowledge the fruitful discussions withSabyasachi Barik and Surya N Sahoo regarding theheterodyne detection technique. We also thank SushreeS Sahoo for assisting in performing the experiment. Thisexperiment was financially supported by the Departmentof Atomic Energy, Govt. of India. [1] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote,and M. D. Lukin, Phys. Rev. Lett e tan, C. Evellin, J. Wolters, Y. Miroshny-chenko, P. Grangier, and A. Browaeys, Phys. Rev. Lett.
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I/I sat = 2 | Ω p | / Γ . For Rb, I sat = 1 .
64 mW and Γ = 2 π ×
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