A Weakly-Interacting Many-Body System of Rydberg Polaritons Based on Electromagnetically Induced Transparency
Bongjune Kim, Ko-Tang Chen, Shih-Si Hsiao, Sheng-Yang Wang, Kai-Bo Li, Julius Ruseckas, Gediminas Juzeliunas, Teodora Kirova, Marcis Auzinsh, Ying-Cheng Chen, Yong-Fan Chen, Ite A. Yu
AA Weakly-Interacting Many-Body System of Rydberg Polaritons Based onElectromagnetically Induced Transparency
Bongjune Kim , Ko-Tang Chen , Shih-Si Hsiao , Sheng-Yang Wang , Kai-Bo Li , Julius Ruseckas , GediminasJuzeli¯unas , Teodora Kirova , Marcis Auzinsh , Ying-Cheng Chen , , Yong-Fan Chen , , and Ite A. Yu , , ∗ Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan Institute of Theoretical Physics and Astronomy,Vilnius University, Saul˙etekio 3, 10257 Vilnius, Lithuania Institute of Atomic Physics and Spectroscopy, University of Latvia, LV-1004 Riga, Latvia Laser Centre, University of Latvia, LV-1002, Riga, Latvia Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan Center for Quantum Technology, Hsinchu 30013, Taiwan
We proposed utilizing a medium with a high optical depth (OD) and a Rydberg state of lowprincipal quantum number, n , to create a weakly-interacting many-body system of Rydberg po-laritons, based on the effect of electromagnetically induced transparency (EIT). We experimentallyverified the mean field approach to weakly-interacting Rydberg polaritons, and observed the phaseshift and attenuation induced by the dipole-dipole interaction (DDI). The DDI-induced phase shiftor attenuation can be viewed as a consequence of the elastic or inelastic collisions among the Ryd-berg polaritons. Using a weakly-interacting system, we further observed that a larger DDI strengthcaused a width of the momentum distribution of Rydberg polaritons at the exit of the system to be-come notably smaller as compared with that at the entrance. In this study, we took n = 32 and theatomic (or polariton) density of 5 × (or 2 × ) cm − . The observations demonstrate that theelastic collisions are sufficient to drive the thermalization process in this weakly-interacting many-body system. The combination of the µ s-long interaction time due to the high-OD EIT mediumand the µ m -size collision cross section due to the DDI suggests a new and feasible platform for theBose-Einstein condensation of the Rydberg polaritons. Rydberg-state atoms provide a strong dipole-dipole in-teraction (DDI) and a dipole blockade mechanism [1–4]suitable for applications in quantum information process-ing, such as quantum simulators [5–7], quantum logicgates [8–11], and single-photon sources [12–14]. The ef-fect of electromagnetically induced transparency (EIT)reduces the propagation speed of light in media and sig-nificantly increases the light-matter interaction time [15–19]. Thus, the EIT effect involving the Rydberg atomscan efficiently mediate significant photon-photon inter-actions [20–25], and this issue has attracted a great dealof attention recently. Most of the previous Rydberg-EIT studies concentrated on strongly-correlated photonsin the dipole blockade regime, which emerges for thestrongly-interacting Rydberg atoms with a high princi-pal quantum number, n , close to 100. This leads to, forexample, the two-body photon-photon gates [11, 24] andstrongly-correlated many-body phases [22, 25].Here we proposed and experimentally demonstrated aninnovative idea of utilizing a medium with a high opti-cal depth (OD) and Rydberg atoms with a low principalquantum number to create a weakly-interacting many-body system based on the EIT effect. In this work, OD ≈
80 and n = 32 were used. We studied the EIT po-laritons also known as the dark-state polaritons [26–28],which, in the case of the Rydberg EIT, are quasi-particlesrepresenting superpositions of photons and Rydberg co-herences. The Rydberg coherence is the coherence be-tween the atomic ground and Rydberg states. Weakly- interacting Rydberg polaritons under an OD-enhancedinteraction time can be employed in the study of many-body physics, such as the Bose-Einstein condensation(BEC) [29–36].The system of the Rydberg EIT polaritons is an en-semble of bosonic quasiparticles [27, 28]. The dispersionrelation and momentum distribution of the Rydberg po-laritons can be associated with their effective mass andthe temperature [36]. The phase shift and attenuationof the output light induced by the DDI can be viewedas a consequence of the elastic and inelastic collisionsamong the polaritons. Hence, the change rates of thephase shift and attenuation are related to the elastic andinelastic collision rates, respectively. Finally, the prop-agation time of slow light or the diffusion time of sta-tionary light in the EIT system is exactly the interactiontime of the particles.The high-OD medium provides for the Rydberg polari-tons a long interaction time of a few µ s, as demonstratedhere and in our previous works [17, 18]. On the otherhand, the collision cross section of polaritons due to theDDI is around µ m in a low- n Rydberg state, leading tothe elastic collision rate of a few MHz at the Rydberg-polariton density of 2 × cm − used in this work. Thelow- n Rydberg state, together with a moderate polari-ton density, ensured a weakly-interacting regime in ourexperiment, where the mean distance between Rydberg-state atoms was much larger than the blockade radius.This made it possible to avoid the severe loss of Rydberg a r X i v : . [ phy s i c s . a t o m - ph ] J un PMTAOM
EMCCDDM QWP BB PD PMF BS b CL b CL a Cold
Atoms MMF L3 M b L2 L1 CL c BS a PBS M a M a QWP DM (b) c p c p |2 |3 |1 (a) FIG. 1: (a) Relevant energy levels and laser excitations in theexperiment. States | i , | i and | i correspond to the groundstate | S / , F = 2 , m F = 2 i , Rydberg state | D / , m J =5 / i , and excited state | P / , F = 3 , m F = 3 i of Rb atoms.Ω c and Ω p denote the coupling and probe Rabi frequencies,and ∆ c and ∆ p represent their detunings. (b) Experimentalsetup. AOM: acousto-optic modulator; BS a , BS b : beam split-ters; CL a , CL b , CL c : collimation lenses; PMF: polarization-maintained optical fiber; BB: beam block; M a : mirror; M b :movable mirror on a flip mount; PD: photo detector (ThorlabsAPD110A); PBS: polarizing beam splitter; DM: dichroic mir-ror; L1, L2, L3: lenses; QWP: quarter-wave plate; EMCCD:electron-multiplying CCD camera (Andor DL-604M-OEM);MMF: multimode optical fiber; PMT: photomultiplier tubeand amplifier (Hamamatsu H6780-20 and C9663). Blue, red,and brown arrowed lines indicate the optical paths of the cou-pling, probe, and AOM’s zeroth-order beams, respectively. polaritons due to the blockade effect. In an ensemblewith a weakly-interacting nature, a high OD can resultin a sufficiently large product of the interaction time andcollision rate for Rydberg polaritons, enabling them toexhibit many-body phenomena [36–38]. Up to now, theBECs of various kinds of polaritons have all been realizedin cavity systems [32–35]. The present work deals withthe cavity-free high-OD medium, in which the interac-tion time between the Rydberg polaritons is analogousto the storage time of polaritons in a cavity with a Q factor greater than 10 .In Ref. [39], we developed a mean field theory to de-scribe weakly-interacting Rydberg polaritons and pre-dicted the DDI-induced phase shift and attenuation.In the present experiment we employed a laser-cooledatomic cloud with an OD of 80, and systematically stud-ied the phase shift and attenuation of an output probelight for the EIT involving a Rydberg state of | D / i .Figure 1(a) shows the transitions driven by the probeand the coupling fields in the EIT system. The experi-mental data are in a good agreement with the theoreticalpredictions, revealing that the low- n Rydberg polaritonsin the high-OD medium can form a weakly-interactingmany-body system.Furthermore, we varied the DDI strength via the in-put photon flux and measured the transverse momentumdistribution of the Rydberg polaritons. A larger DDIstrength caused the width of the momentum distributionto become notably smaller, indicating the thermalizationprocess was driven by elastic collisions [36, 40, 41]. Theobserved reduction of the momentum distribution width suggested that the combination of the µ m-range DDIstrength and the µ s-long light-matter interaction timecan make the BEC of the Rydberg polaritons feasible.The mean field model developed in Ref. [39] is sum-marized in this paragraph. Rydberg excitations are ran-domly distributed and can be considered approximatelyas particles of an ideal gas due to their weak interaction.Thus, the nearest-neighbor distribution (NND) [42] wasutilized in our theory. Using the probability function ofthe NND and the atom-light coupling equations of anEIT system, we derived the following analytical formulasof the steady-state DDI-induced attenuation coefficient∆ β and phase shift ∆ φ :∆ β = 2 S DDI (cid:18) √ W c − c W c − γ √ W c + 2∆ c Ω c + 3 δ √ W c − c Ω c (cid:19) Ω p , (1)∆ φ = ± S DDI (cid:18) √ W c + 2∆ c W c − γ √ W c − c Ω c − δ √ W c + 2∆ c Ω c (cid:19) Ω p , (2) S DDI ≡ π α Γ p | C | n atom ε c , (3) W c ≡ p Γ + 4∆ c , (4)where α is the OD of the system, Γ is the spontaneousdecay rate of excited state | i , C is the van der Waalscoefficient in SI units of Hz · m , γ is the intrinsic de-coherence rate of the experiment, δ = ∆ p + ∆ c is thetwo-photon detuning, n atom is the atomic density, and ε is a phenomenological parameter. Utilizing ε , we re-lated the average Rydberg-state population ( ρ ) to theinput probe and coupling Rabi frequencies (Ω p and Ω c )as ρ = ε Ω p / Ω c . In Eq. (2), the sign of + indicates C < − indicates C >
0. To reach theabove formulas, we assumed Ω p (cid:28) Ω c and r B (cid:28) r a ,where r B is the blockade radius and r a is the half meandistance between Rydberg polaritons.The attenuation coefficient β and phase shift φ of theprobe field are defined by the output-input (Ω p -Ω p ) rela-tion of Ω p = Ω p exp( iφ − β/ β = β + ∆ β ≈ αγ ΓΩ c + ∆ β, (5) φ = φ + ∆ φ ≈ α Γ δ Ω c + ∆ φ, (6)where β and φ are the attenuation coefficient and phaseshift in the absence of DDI [39]. The mean field theorypredicted that both β and φ depend on Ω p linearly, andthe dependence of Ω p comes from the Rydberg-state pop-ulation ρ . In addition, the slope of β or φ versus Ω p isasymmetric with respect to ∆ c = 0.We carried out the experiment in cold Rb atoms thathad been produced by a magneto-optical trap (MOT).There were typically 5 × trapped atoms with a tem-perature of about 350 µ K in the MOT [43, 44]. Be-fore each measurement, we optically pumped all popu-lation to a single Zeeman state | F = 2 , m F = 2 i at theground level | S / i [45]. Since the probe and couplingfields were σ + -polarized, each of the energy levels | i , | i and | i shown in Fig. 1(a) was a single Zeeman state.The spontaneous decay rate, Γ, of | i is 2 π × | i ↔ | i is a cycling transition. The life time,Γ − , of | i is about 30 µ s [46], and Γ (cid:28) γ in thiswork. Thus, the influence of Γ was negligible. We es-timated n atom ≈ × cm − in the experiment, and C = − π ×
260 MHz · µ m for | D / , m J = 5 / i by us-ing D ϕ = 0.343 [47]. Such C together with n atom canmake the DDI effect unobservable under a small Ω p or alow OD [48–51].Figure 1(b) shows the experimental setup. The probeand coupling fields counter-propagated along the majoraxis of the atom cloud and completely overlapped witheach other. The e − full width of the probe and couplingbeams were 130 and 250 µ m, respectively. Details of theexperimental setup can be found in Sec. I of the Sup-plemental Material. The entire setup of red and brownoptical paths shown in Fig. 1(b) formed the beat-noteinterferometer, which measured the phase shift of theoutput probe field. Details of the beat-note interferom-eter and the phase measurement can be found in Sec. IIof the Supplemental Material and our previous work inRefs. [52, 53]. The brown optical paths were present onlyin the phase shift measurement.The parameters of Ω c , α (OD), and γ were determinedexperimentally with the same method used in Ref. [43].Details of the determination procedure can be found inSec. III of the Supplemental Material. We set Ω c = 1 . γ around 0.012(1)Γ through-out the experiment. This γ included the effects of laserfrequency fluctuation, Doppler shift, and other decoher-ence processes that appear in the Λ-type EIT system [43].To verify the mean field theory of weakly-interactingRydberg polaritons, we measured the attenuation coef-ficient β and phase shift φ as functions of Ω p , as shownin Figs. 2(a) and 2(c). A nonzero two-photon detuning, δ , can significantly affect the phase shift and attenua-tion of the output probe light in the EIT medium. Thus,we utilized a beat-note interferometer to carefully deter-mine the probe frequency for δ = 0. The determinationprocedure is illustrated in Sec. II of the SupplementalMaterial. The uncertainty of δ/ π was ±
30 kHz, includ-ing the accuracy of the beat-note interferometer of ± ±
20 kHz.At δ = 0, we applied a square input probe pulse andmeasured the steady-state attenuation coefficient β andphase shift φ of the output probe pulse. The measure- (r a d i a n s ) p (units of 10 -2 ) S l op e o f ( un it s o f - ) -2 0 202040 c (units of ) S l op e o f ( un it s o f - ) (a) (b)(c) (d) FIG. 2: (a,c) Attenuation coefficient β and phase shift φ at δ = 0 as functions of Ω p in the presence of the DDI. Circlesare the experimental data taken with OD = 81 ±
3, Ω c = 1 . c = −
2Γ (black), −
1Γ (red), 0 (blue), 1Γ (magenta),and 2Γ (olive), where Γ = 2 π × c . Black lines are thebest fits, which determine S DDI = 37/Γ / in (b) and 38/Γ / in (d). ment procedure and representative data can be found inSec. IV of the Supplemental Material. Figure 2(a) [and2(c)] shows β (and φ ) versus Ω p at various ∆ c , where Ω p corresponds to the peak or center intensity of the inputGaussian beam. At a given ∆ c , the data points approxi-mately formed a straight line, as expected from the the-ory. We fitted the data with a linear function and plottedthe slope of the best fit against ∆ c , as shown in Fig. 2(b)[and 2(d)]. It can be noticed that the y -axis interceptionsof best fits were scattered in Fig. 2(c). Since a change of30 kHz in δ/ π resulted in that of 0.4 rad in phase, theuncertainty of δ made the interceptions scatter aroundzero.The strengths of the DDI effect on β and φ , i.e., theslope of β versus Ω p and that of φ versus Ω p (denoted as χ β and χ φ ), were asymmetric with respect to ∆ c = 0 asshown in Figs. 2(b) and 2(d). Such asymmetries are ex-pected from the theory, as demonstrated by Eqs. (1) and(2). Considering the coupling detunings of −| ∆ c | and | ∆ c | , χ β (or χ φ ) of −| ∆ c | was always larger (or smaller)than that of | ∆ c | . The physical picture of the asymme-tries is discussed in Ref. [39]. We fitted the data pointsin Figs. 2(b) and 2(d) with the functions of χ β and χ φ ,where δ and γ were set to 0 and 0.012Γ, and S DDI wasthe only fitting parameter. The S DDI of the best fit shownin Fig. 2(b) is in good agreement with that of Fig. 2(d).Using Eq. (3) and the values of S DDI , C , n atom and Ω c ,we obtained ε = 0.43 ± ε relatedthe average Rydberg-state population to Ω p and that Ω p ( f =250mm) L4 L5 L3L2 ( f =200mm) ( f =75mm)( f = CL b Atoms
FIG. 3: Sketch of key elements for images of probe beam onthe EMCCD. Values of f indicate the focal lengths of lenses.PMF, CL b , L2 and L3 here are the same as those in Fig. 1(b).The probe beam coming out of CL b was collimated and hadthe e − full width of 0.92 mm. We adjusted the separationbetween L4 and L5 such that the beam was focused on nearlythe center of the atomic cloud by L2, and had the width of39 µ m at the focal point. corresponded to the center intensity of input Gaussianbeam, the measured value of S DDI is also reasonable.Thus, the experimental data verified the mean field the-ory of weakly-interacting Rydberg polaritons.To observe the thermalization process in this weakly-interacting system, we varied the DDI strength and mea-sured the transverse momentum distribution of the Ry-dberg polaritons. The moveable mirror M b shown inFig. 1(b) was installed to direct the output probe beamto the EMCCD. Figure 3 shows the key elements for theEMCCD image of the probe beam. We reduced the waistof the input probe beam by adding the lens pair of L4 andL5, as shown in the figure, and made the input photonsor initial Rydberg polaritons have a large transverse mo-mentum distribution. The strengths of the DDI effect on β and φ (i.e., χ β and χ φ ) inferred the inelastic and elas-tic scattering rates of Rydberg polaritons in the medium,respectively. The elastic collisions produced the favoredprocess of thermalization, while the inelastic collisionscaused the disfavored process of decay. Hence, we chose∆ c = 1Γ to have a suitable ratio of elastic to inelasticscattering rates for the study.The elastic collision rate was estimated with the for-mula of phase shift per unit time, dφ/dt , being equalto the phase shift per collision, φ c , multiplied by thecollision rate, R c . Using the propagation delay timeand the data point shown in Fig. 2(c) of ∆ c = 1Γ andΩ p = 0 .
2Γ resulted in dφ/dt = 0.64 rad/ µ s. Note thatthe experimental condition of Ω p = 0 .
2Γ correspondedto the Rydberg-polariton density of 2 × cm − . Con-sidering the hard-sphere collision, φ c = ¯ ka where ~ ¯ k isthe root-mean-square value of the relative momentumbetween two colliding bodies, and a is the radius of ahard sphere. In view of the Rydberg polaritons, a can betreated as the blockade radius [54], which was about 2.1 µ m in our case. The momentum distribution shown inFig. 4(d) indicated ¯ k = 0 . µ m − . Thus, φ c = 0.16 rad,inferring R c = 4.0 MHz. Under such an elastic collisionrate, it was feasible to observe the thermalization effectin our experiment.We measured the output probe beam size on the EM- (b) (a) x y (c) (d) (e) k x k y (f) -1 FIG. 4: (a-c) Images of the probe beam profile taken by theEMCCD at Ω p = 0 .
2Γ in the absence of the atoms, and atΩ p = 0 .
1Γ and 0 .
2Γ in the presence of the atoms, respectively.Color represents the gray level detected by the EMCCD. Wefit each image with a Gaussian function, and draw a white-dashed-line circle of the diameter equal to the e − full widthof the best fit. The diameters of the circles from top to bot-tom are 2.6, 2.2 and 1.6 mm. (d-f) Transverse momentumdistributions of the probe photons at the output, i.e., the Ry-dberg polaritons in the atomic cloud, derived from the bestfits of the images in (a-c), respectively. Color represents thenormalized probability density. The e − full widths of the dis-tributions from top to bottom are 0.15, 0.12 and 0.092 µ m − . CCD at ∆ c = 1Γ. To avoid the lensing effect [55], weset the two photon detuning, δ , corresponding to thezero phase shift, i.e., φ = 0, in the measurement. Pleasenote φ = 0 is the condition that the phase shift due tothe ordinary EIT effect at δ cancels out the phase shiftdue to the DDI effect. Details of the measurement of theprobe beam size, i.e., the transverse momentum distribu-tion of the Rydberg polaritons, and those of the study onthe lensing effect can be found in Secs. V and VI of theSupplemental Material. The image of the probe beam,in the absence of the atoms, is shown in Fig. 4(a), whilethe two images in the presence of the atoms are shownin Figs. 4(b) and 4(c). It can be clearly observed that alarger value of Ω p , i.e., a higher density of Rydberg po-laritons because of n atom ρ ∝ Ω p , caused the beam sizeto become smaller. Since transverse momentums of theRydberg polaritons were carried by the probe photonsleaving the medium, the intensity profile of the outputprobe beam was able to be used to derive the transversemomentum distribution [32, 36]. We fitted the intensityprofiles of three images with a two-dimension Gaussianfunction, and utilized the results of the best fits to con-struct the momentum distributions shown in Figs. 4(d)-4(f). As the Rydberg-polariton density increased, theelastic collision rate also increased. Due to Ω c = 1.0Γ,the values of Ω p in the cases of Figs. 4(b) and 4(c) wellsatisfied the perturbation condition. Thus, the propaga-tion times of the probe light, i.e., the interaction timesof the Rydberg polaritons, were approximately the samein the two cases. Under the same interaction time, thehigher collision rate due to the larger Rydberg-polaritondensity produced a smaller width of the transverse mo-mentum distribution or a lower effective transverse tem-perature, which is the expected outcome of the thermal-ization process.In conclusion, we proposed utilizing a high-OD EITmedium and a low- n Rydberg state to create a weakly-interacting system of Rydberg polaritons, which can bea platform for the study of many-body physics, such asthe polariton BEC. The experimental data of the atten-uation coefficient and the phase shift influenced by theDDI are consistent with the predictions of the mean fieldtheory developed in Ref. [39]. Furthermore, we measuredthe transverse momentum distribution of the Rydbergpolaritons in the medium. A larger probe intensity atthe input, i.e., a larger Rydberg-polariton density or ahigher elastic collision rate, resulted in a smaller probeintensity profile at the output, i.e., a smaller momen-tum distribution width or a lower effective temperatureof the Rydberg polaritons. The thermalization processwas observed in this weakly-interacting many-body sys-tem. The DDI strength can be controlled and easily var-ied by the principal quantum number of the Rydbergstate, the input probe intensity, and the coupling detun-ing and intensity in the EIT system, offering degrees offreedom in experiments. Our proposal is supported bythe experimental demonstration and opens a new avenuein the study of quasi-particles in the field of many-bodyphysics.
Acknowledgments
This work was supported by Grant Nos. 105-2923-M-007-002-MY3, 107-2745-M-007-001, and 108-2639-M-007-001-ASP of the Ministry of Science and Technol-ogy of Taiwan, Project No. TAP LLT-2/2016 of the Re-search Council of Lithuania, and Project No. LV-LT-TW/2018/7 of the Ministry of Education and Scienceof Latvia. JR and GJ also acknowledge a support by theNational Center for Theoretical Sciences, Taiwan. ∗ Electronic address: [email protected][1] M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D.Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. ,037901 (2001). [2] D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y.P. Zhang, R. Cˆot´e, E. E. Eyler, and P. L. Gould, Phys.Rev. Lett. , 063001 (2004).[3] R. Heidemann, U. Raitzsch, V. Bendkowsky, B. Butscher,R. L¨ow, L. Santos, and T. Pfau, Phys. Rev. Lett. ,163601 (2007).[4] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod.Phys., , 2313 (2010).[5] H. Weimer, M. M¨uller, I. Lesanovsky, P. Zoller, and H.P. B¨uchler, Nat. 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408 (1996).
UPPLEMENTAL MATERIALA Weakly-Interacting Many-Body System of Rydberg Polaritons Based onElectromagnetically Induced Transparency
Bongjune Kim , Ko-Tang Chen , Shih-Si Hsiao , Sheng-Yang Wang , Kai-Bo Li , Julius Ruseckas , GediminasJuzeli¯unas , Teodora Kirova , Marcis Auzinsh , Ying-Cheng Chen , , Yong-Fan Chen , , and Ite A. Yu , , ∗ Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan Institute of Theoretical Physics and Astronomy,Vilnius University, Saul˙etekio 3, 10257 Vilnius, Lithuania Institute of Atomic Physics and Spectroscopy, University of Latvia, LV-1586 Riga, Latvia Laser Centre, University of Latvia, LV-1002, Riga, Latvia Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan Center for Quantum Technology, Hsinchu 30013, Taiwan
I. EXPERIMENTAL SETUP
In our experiment of the Rydberg-state electromagnet-ically induced transparency (EIT), the probe and cou-pling fields were generated by a homemade diode laserand a blue laser system (Toptica TA-SHG pro), respec-tively. The frequency stabilizations of the probe and cou-pling lasers are described as follows. We utilized theinjection lock scheme to stabilize the frequency of theprobe laser. In the scheme, the probe laser was usedas the slave, and an external-cavity diode laser (ECDL)of Toptica DLC DL pro with the wavelength of about780 nm was employed as the master. We utilized thePound-Drever-Hall (PDH) scheme and the saturated ab-sorption spectroscopy to lock the frequency of the ECDLwith a heated vapor cell of Rb atoms. The blue laserhad the wavelength of about 482 nm. We utilized thePDH scheme and the EIT spectroscopy to lock the fre-quency of the blue laser with another heated vapor cell.In this EIT spectroscopy, the ECDL and blue laser fieldscounter-propagated. We were able to lock the sum ofthe two laser frequencies to the EIT transition with theroot-mean-square fluctuation of around 150 kHz [1].As shown by the red path in Fig. 1(b) of the maintext, the probe field came from the first-order beam ofan acousto-optic modulator (AOM) and was coupled to apolarization maintained fiber (PMF). The PMF deliveredthe probe field to the cold atoms. Not shown in the figure,the probe field was twice diffracted from the AOM in thedouble-pass scheme. The AOM was used to shape theprobe pulse, and its driving rf frequency and amplitudewere precisely tuned or varied by a function generator.We used a rubidium atomic standard (SRS FS752) as theexternal clock of the function generator. The blue path inFig. 1(b) of the main text represents the coupling field,which was switched on and off with another AOM not ∗ Electronic address: [email protected] shown in the figure. Inside the atom cloud, the probeand coupling fields counter-propagated and completelyoverlapped to minimize the Doppler effect. The e − fullwidths of the probe and coupling beams were 130 and250 µ m, respectively.Before each measurement, we first turned off the mag-netic field of the magnetic-optical trap (MOT) and thenperformed the dark MOT for 2.5 ms to increase the op-tical depth (OD) of the system [2, 3]. When the experi-mental condition of a low OD was needed, we were ableto adjust the dark-MOT parameter to reduce the ODof the system by about 4 folds. The cigar-shaped atomcloud had the dimension of 1.8 × × [4]. Theprobe and coupling fields propagated along the majoraxis of the atom cloud. After passing through the atoms,the probe field was detected by a photomultiplier tube(PMT). A digital oscilloscope (Agilent MSO6014A) ac-quired the signal from the PMT and produced raw data. II. BEAT-NOTE INTERFEROMETER FOR THEPHASE MEASUREMENT
We employed the beat-note interferometer to measurethe phase shift of the probe field induced by the atoms.The concept and illustration of the beat-note interfer-ometer can be found in Refs. [5, 6]. In the absence ofthe dipole-dipole interaction (DDI), the phase shift ofthe probe field also enabled us to precisely determine theprobe frequency for the zero two-photon detuning, i.e., δ = 0, at a given coupling detuning, ∆ c . This is becausethe phase shift is equal to τ d δ according to the non-DDIEIT theory, where τ d is the propagation delay time. Thedata of attenuation coefficients and phase shifts underthe DDI effect presented in Fig. 2 of the main text weretaken at δ = 0.As shown by the brown path in Fig. 1(b) of the maintext, the zeroth-order beam of the AOM was unblockedwhen we performed the phase measurement. The first-order beam of the AOM was the probe field. We com-bined the probe beam and the zeroth-order beam with a r X i v : . [ phy s i c s . a t o m - ph ] J un N o r m a li ze d B ea t N o t e ( a r b . un it s ) Time (ns) B ea t N o t e ( m V ) Time ( s) (a) (b) P h a s e D i ff e r e n ce (r a d ) Time ( s) (c) × FIG. S1: Determination of the phase shift with the beat-note interferometer. (a) Representative data of the probe and referencebeat notes as functions of time are shown by red and black lines. The high-frequency component of the lines is the beat note.The black line is scaled down by a factor of 0.2. (b) We zoom in the data around 5.2 µ s of (a) to demonstrate the phasedifference between probe (red circles) and the reference (black circles) beat notes. Red and black lines are the best fits ofsinusoidal functions. (c) Phase difference measured with the atoms (red circles) and that measured without the atoms (bluecircles) as functions of time. The two measurements were taken consecutively. Red line is the exponential best fit of the redcircles and blue line is the average value of the blue circles. The experimental data were measured at OD = 80, Ω c = 1 . c = 0, δ = 0, and Ω p = 0 . a 50/50 beam splitter, BS a , in the figure to form thebeat note, which had a fixed and stable frequency setby the AOM’s driving frequency. The PMF can ensureboth beams to completely overlap and have the samepolarization. Coming out of the PMF, the beat-notesignal was split into two by another 50/50 beam split-ter, BS b . One, named the reference beat note, did notpass through the atoms and was detected by a photodetector (PD). The other, name the probe beat note,passed through the atoms and was detected by the PMT.Both beat notes were measured simultaneously. We com-pared the PMT and PD output signals to determine thephase difference between the probe and reference beatnotes. Since the beat-note frequency (or wavelength)was around 133 MHz (or 2.3 m), the phase differencemeasured by the beat note interferometer was insensi-tive to the change or flucutation of the optical path [5].The insensitivity was demonstrated by the result that thephase difference measured without the atoms was a sta-ble constant. In addition, the frequency of the zero-orderbeam was far away from the transition frequency. Thus,the measured phase difference was mainly contributedfrom the phase shift of the probe field. We subtractedthe phase difference measured with the atoms from thatmeasured without the atoms to obtain the phase shift ofthe probe field induced by the atoms.We carefully set the two-photon detuning to zero at agiven coupling detuning (∆ c ), before measuring the ad-ditional phase shift and attenuation induced by the DDI.To determine the probe frequency for δ = 0, the experi-mental condition was changed to α (OD) = 22 ∼
26 andΩ p ≈ π × × samples persecond and stored in the segmented memory by the os-cilloscope. We averaged beat-note data for 936 times anddetermined the phase difference between the probe andreference beat notes with the sinusoidal best fits of thedata. For example, the high-frequency components of thelines in Fig. S1(a) and the circles in Fig. S1(b) demon-strate the beat-note data, while the red and black linesin Fig. S1(b) are the best fits. Once obtaining the probefrequency for δ = 0, we immediately switched back to theexperimental condition of OD ≈
81 and Ω p ≥ . III. DETERMINATION OF EXPERIMENTALPARAMETERS
The experimental parameters of the coupling Rabi fre-quency, Ω c , the optical depth (OD), α , and the intrin-sic decoherence rate, γ , were determined in the samemethods described in Ref. [1]. As the examples of thedetermination methods, Figs. 7(a)-7(c) of this referencedemonstrated the data of a similar Rydberg-EIT experi-ment.We determined Ω c by the frequency separation of twominima, i.e. the Autler-Towns splitting, in the probetransmission spectrum measured at the coupling detun-ing, ∆ c , of nearly zero and OD ≈
1. An example of thespectrum is shown in Fig. 7(a) of Ref. [1]. Suppose thefrequencies of the two minima are denoted as δ − and δ + .Then, Ω c was obtained by the relation of δ + − δ − ≈ Ω c + ∆ c c (S1)under the condition of ∆ c (cid:28) Ω c . The difference betweenthe magnitudes of δ − and δ + is given by | δ + | − | δ − | = ∆ c . (S2)Using the above difference, we carefully minimized | ∆ c | inthe measurement such that the correction term ∆ c / c in Eq. (S1) was about 1 . × − Γ. The condition of ∆ c =0 was also determined in this step. To obtain the Autler-Towns splitting, we swept the probe frequency by varyingthe driving frequency of the AOM shown in Fig. 1(b) ofthe main text. The double-pass scheme of the AOM madethe input probe power relatively stable during each runof the frequency sweeping. To prevent the spectrum frombeing distorted by the transient effect, a sufficiently slowsweeping rate of 240 kHz/ µ s was utilized.After the value of Ω c was known, we determined theOD, i.e., α , in a designated study by measuring the prop-agation delay time of the probe pulse. An example ofthe delay time measurement is shown in Fig. 7(b) ofRef. [1]. A short Gaussian input pulse with e − full widthof 0.66 µ s was used in the measurement. To minimizethe DDI effect, we employed a weak probe pulse with thepeak Rabi frequency of 0.05Γ. According to the non-DDIEIT theory, the delay time, τ d is given by τ d = α ΓΩ c . (S3)We used the measured value of τ d to obtain the OD ofthe experiment in the designated study.The intrinsic decoherence rate γ was determined bymeasuring the probe transmission at both of the two-photon and one-photon resonances, i.e., not only δ = 0but also ∆ c = 0. In the measurement, we set Ω c = 1.0Γand α = 24. The input Gaussian pulse of the probefield had the e − full width of 7 µ s and the peak Rabifrequency of 0.05Γ. The low OD and the weak probepulse were used to made the DDI effect negligible. Thelong probe pulse made the measured transmission as thesteady-state result. According to the non-DDI EIT the-ory, the peak transmission T max , i.e., the transmission at δ = 0, is given by T max = exp (cid:18) − α ΓΩ c γ (cid:19) . (S4)The measured value of T max inferred the intrinsic deco-herence rate of the experiment. We determined γ beforeeach study and its values maintained at (11 ∼ × − Γthroughout this work. T r a n s m i ss i on ( % ) Time ( s) × FIG. S2: Representative data of the probe transmission asa function of time taken at OD = 80, Ω c = 1 . c = 0, δ = 0, and Ω p = 0 . µ s to calculate the attenuationcoefficient. IV. MEASUREMENTS OF TRANSMISSIONAND PHASE SHIFT OF THE OUTPUT PROBEFIELD
In this section, we will describe the measurement pro-cedure of the probe transmission and phase shift underthe DDI effect. To make the DDI effect significant, a highOD of 81 ± c of (1.0 ± ≤ Ω p ≤ c , such weak Ω p makes the probe field as the perturba-tion. Under the perturbation limit and without any DDIeffect, the attenuation coefficient and phase shift of theoutput probe field do not change by varying Ω p .Figures 2(a) and 2(c) of the main text show the attenu-ation coefficient and phase shift of the output probe fieldas functions of Ω p . There are five different values of thecoupling detuning, ∆ c , in each figure. We performed aset of consecutive measurements at each value of ∆ c . Ineach measurement set, we carried out the following steps:(i) The experimental parameters of Ω c , α , and γ wereverified by the methods described in Sec. III. (ii) We setand checked a designated value of ∆ c . (iii) The probefrequency for the two-photon resonance condition, i.e., δ = 0, was searched for and determined by the beat-noteinterferometer illustrated in Sec. II. (iv) We measured theprobe transmissions or phase shifts at different values ofΩ p consecutively. (v) The steps (i), (ii), and (iii) wereperformed in the reverse order to ensure that the experi-mental condition did not change. Since it took about onehour to complete the steps (iii) and (iv), we confirmedthe two-photon frequency had an uncertainty of merely ±
20 kHz due to the long-term drift. Because of the abovesteps, it can be certain that the additional attenuationand phase shift were contributed mainly from the varia-tion of Ω p , i.e. from the DDI effect, but very little fromthe fluctuation or change of δ and γ .The representative data of the input and output probetransmissions are shown in Fig. S2. The data were av-eraged 512 times by the oscilloscope. We calculated thelogarithm of the mean output transmission in the timeinterval between 5 and 6 µ s to obtain the value of − β ,where β is the attenuation coefficient. The attenuationcoefficients at different values of Ω p and ∆ c shown inFig. 2(a) of the main text were determined in the sameway. The representative data of the reference and probebeat notes are shown in Fig. S1(a) and illustrated inSec. II. The data look like some high-frequency compo-nents added to the signals shown in Fig. S2. These high-frequency components are the beat notes. We zoomedin each time interval of 50 ns, and determined the phasedifference between the probe and reference beat notes. Inthe absence of the atoms, the phase difference was a sta-ble constant of 0.71 radians as shown by the blue circlesin Fig. S1(c). In the presence of the atoms, the represen-tative data of phase difference as a function of time areshown by the red circles in Fig. S1(c). The red line inthe figure is the best fit of the red circles. We subtractedthe steady-state value of the red line from 0.71 radiansto obtain the phase shift of the probe field induced bythe atoms. The phase shifts at different values of Ω p and∆ c shown in Fig. 2(c) of the main text were determinedin the same way. V. MEASUREMENT OF TRANSVERSEMOMENTUM DISTRIBUTION OF RYDBERGPOLARITONS
To observe the evidence of the DDI-induced thermal-ization process in the Rydberg-polariton system, we stud-ied the transverse momentum distribution of the Rydbergpolaritons by taking images of the probe beam. Afterentering the medium of the atoms, the transverse mo-mentums of the input probe photons were converted tothose of the Rydberg polaritons. In the medium, theDDI can induce the elastic collisions among the Rydbergpolaritons and reduce the width of their transverse mo-mentum distribution. As the probe beam was exiting themedium, the transverse momentums of the Rydberg po-laritons were converted back to those of the output pho-tons. The images of the input and output probe beamprofiles revealed the initial and final momentum distri-butions of the Rydberg polaritons. It can be seen as theconsequence of cooling after the thermalization processthat the output probe beam size becomes smaller thanthe input one, i.e., the width of the final transverse mo-mentum distribution becomes narrower than that of theinitial one. However, the lensing effect of the atoms canalso change the probe beam size. The condition of thezero phase shift, i.e. φ = 0 can avoid the lensing effect,which will be explained in Sec. VI. For each of differentvalues of Ω p , we first searched for the two-photon detun-ing, δ , corresponding to φ = 0, and then took the imageof the output probe beam profile at δ . p (units of 10 -2 ) FIG. S3: Attenuation coefficient, β , as a function of Ω p inthe case II at OD = 82, Ω c = 1.0Γ, ∆ c = 1.0Γ, and δ = 0.Circles are the experimental data and straight line is the bestfit. The slope, χ β , of the best fit is (16 ± / Γ here, while χ β is (17 ± / Γ in the case I, measured at OD = 81, Ω c = 1.0Γ,∆ c = 1.0Γ, and δ = 0. To clearly observe the thermalization effect, we madethe Rydberg polaritons have a large initial transverse mo-mentum distribution. This was achieved by reducing thewaist of the input probe beam by adding the lens pair ofL4 and L5 shown in Fig. 3 of the main text. With (andwithout) the lens pair, the e − full width of the probebeam at the position of atom cloud center was 39 µ m(and 130 µ m). As shown in Fig. 1(b) of the main text,the moveable mirror M b directed the output probe beamto the electron-multiplying charge-coupled device (EM-CCD) camera instead of to the PMT. Let’s call the ex-perimental system without (and with) the installation ofL4, L5, and M b as case I (and case II). The experimentalparameters of Ω c and OD of the two cases under the samesettings differed a little, i.e., less than 2%. We tuned thesettings slightly to make the measured Ω c and OD of thetwo cases become the same. Then, the measured deco-herence rates, γ , had no observable difference. It tookabout a minute to switch from one case to another. Weswitched back and forth between the two cases in somemeasurements, and the experimental parameters of eachcase under the same settings were unchanged throughoutthe day.Using nearly the same Ω c , OD, and γ in the measure-ments of the two cases, we also confirmed that the slope, χ β , of the attenuation coefficient, β , versus Ω p in the caseI was consistent with that in the case II. Figure S3 shows β as a function of Ω p which was measured in the caseII at the zero two-photon detuning, i.e. δ = 0. In themeasurement, the input probe field was a Gaussian pulsewith the e − full width of 12 µ s, since we took the EM-CCD image with the similar input pulse. Derived fromthe slope of the straight line in Fig. S3, the value of χ β was (16 ± . According to the data in Fig. 2 of themain text, χ β under nearly the same experimental pa-rameters in the case I was (17 ± . Considering theuncertainties, the difference between the two χ β values isacceptable.In the case II, the probe beam size was significantly re- -300 -200 -100 0 100-2-1012 (r a d i a n s ) /(2 ) (kHz) FIG. S4: Determination of the two-photon detuning, δ , forthe zero phase shift, i.e. φ = 0. Black (or red) circles are theexperimental data of phase shift versus two-photon detuningmeasured at OD = 82, Ω c = 1.0Γ, ∆ c = 1.0Γ, and Ω p = 0.10Γ(or 0.20Γ). We set the probe frequency for δ = 0 using themethod in Sec. II. Black and red lines represent the linear bestfits. We determined δ by the intersection between the best fitand the gray dashed line. As references, blue and green linesare the theoretical predictions calculated numerically [7]. duced. Even with the Rabi frequency of 0.2Γ, the probepower was still too weak to produce reasonable beat-notesignals. Therefore, the beat-note interferometer was notable to provide a sufficient accuracy for the determina-tion of the probe phase shift in the case II. Instead, wesearched for the two-photon detuning, δ correspondingto the zero phase shift, i.e, φ = 0, in the case I. Since thevalues of S DDI of the two cases were consistent, δ foundin the case I can be applied to the case II.The procedure of taking the image of the output probebeam profile is summarized in the following steps: (i)The experimental parameters were determined and set tothe designed values. (ii) The probe frequency for δ = 0was determined in the case I by the method described inSec. II. (iii) At the above experimental parameters, wemeasured S DDI in the case II with the method similar tothat shown in Fig. S3, and confirmed that the measuredvalue was consistent with the value of S DDI in the caseI. (iv) The two-photon detuning, δ , corresponding to φ = 0 was determined in the case I. Representation dataare shown in Fig. S4. We fitted the data with a straightline, and the best fit determined δ . The uncertainty of δ was ± π ×
30 kHz. (v) At δ , we took images of theoutput probe beam profile in the case II. We repeatedthe steps (iv) and (v) for different values of Ω p . (vi) Thevalue of S DDI in the case II was measured again to verifyit was unchanged during the steps (iv) and (v).
VI. STUDY ON THE LENSING EFFECT
In this section, we report the study on the lensing ef-fect of the atoms. We experimentally demonstrated thatcondition of the zero phase shift, i.e., φ = 0, can avoidthe lensing effect. Images of the output probe beam pro-files were taken by the EMCCD for the study. Since the -2 -1 0 1 223 P r ob e B ea m W i d t h on t h e I m a g e ( mm ) (radians) FIG. S5: Demonstration of the lensing effect. We measuredthe e − full width of the probe beam profile on the EMCCDimage in the case II as a function of the probe phase shift.Circles are the experimental data taken at OD = 13, Ω c =1.0Γ, ∆ c = 0, and Ω p = 0.15Γ. Black line is the best fitcalculated with the consideration of the atoms behaving like aGRIN lens. The best fit gives d = 19.5 mm, p = 300 rad · mm ,and z = 1.8 mm, where the meanings of d , p , and z can befound in the text. Gray dashed line is the width measuredwithout the presence of the atoms. beam profile can also be affected by the thermalizationprocess induced by the elastic collisions between the Ry-dberg polaritons, we performed the study with a low ODof 13 to minimize the DDI effect.Figure S5 shows the e − full width of the output probebeam profile as a function of the phase shift. The widthwas measured by the EMCCD image in the case II. Thephase shift was converted from the two-photon detuning, δ , used in the measurement with the following formula: φ = α ΓΩ c δ. (S5)We verified the above formula and determined the probefrequency corresponding to δ with the beat-note interfer-ometer in the case I. When taking images, we employedthe Gaussian probe pulse of the e − full width of 12 µ s.To produce the images of a better signal-to-noise ratio,the peak Rabi frequency of the pulse was 0.15Γ.The gray dashed line in Fig. S5 represents the beamwidth measured without the presence of the atoms. Itcan be clearly seen that the experimental data point of φ = 0 nearly locates on the gray dashed line, indicatingthat the condition of φ = 0 can avoid the lensing effect ofthe atoms. Such a result is also expected from the theoryas shown by the black line in the figure. To calculate theblack line, we utilized the ray-tracing method for Gaus-sian beams. In addition, we considered that the atomsbehaved like a gradient index (GRIN) lens of which therefractive index, n , changes with the transverse distance, r , from the symmetric or central axis in the direction oflight propagation. The ABCD matrix of a GRIN lens is +50 P r ob e B ea m W i d t h on t h e I m a g e ( mm ) /(2 ) (kHz) -50 FIG. S6: Verification of the uncertainty in the two-photondetuning as well as the lensing effect being insignificant inthe measurement of the probe beam width, i.e., the transversemomentum distribution of the Rydberg polaritons. Black andred circles are the experimental data of the e − full width ofthe output probe beam versus the two-photon detuning. Theblack (or red) data points were taken at OD = 82, Ω c = 1.0Γ,∆ c = 0, and Ω p = 0.10Γ (or 0.20Γ). Black and red lines arethe linear best fits of the experimental data, and their slopesare 2.5 µ m/kHz and 2.1 µ m/kHz, respectively. Gray dashedline indicates the probe beam width measured without thepresence of the atoms, i.e., the input probe beam width. given by cos( gd ) sin( gd ) /g − g sin( gd ) cos( gd ) , (S6)where d is the thickness of the lens and g is defined by n ( r ) = n − g r / n the refractive index alongthe central axis [8]. Since the refractive index is propor-tional to the phase shift, φ , of the atoms, the followingrelation is used. g = φp , (S7) where 1 /p is the proportionality. We substituted p φ/p for g in the ABCD matrix to calculate the black line inFig. S5. The black line is the best fit of the experimen-tal data with the fitting parameters of d = 19.5 mm, p = 300 rad · mm , and z = 1.8 mm, where z is the dis-tance between the input beam waist and the center of thelens medium. Thus, the experimental data of the outputprobe beam width as a function of the phase shift canbe well described by the effect of the GRIN lens made ofthe atom cloud.The DDI among the Rydberg polaritons produced aphase shift at δ = 0 and red-shifted the frequency of φ = 0. To avoid the lensing effect of the atoms as wellas to observe the thermalization effect due to the DDI atthe high OD, we searched for the two-photon detuning, δ , corresponding to φ = 0 for each DDI strength or eachvalue of Ω p . The procedure of searching for δ and thentaking the image has been described in Sec. V. However,there was an uncertainty of ± π ×
30 kHz in δ due to theaccuracy of the beat-note interferometer and the long-term drift of the two-photon frequency.To check whether this uncertainty could be significanton the DDI effect, we measured the probe beam size notonly at δ but also at δ ± π ×
50 kHz. Figure S6 showsthe measured beam widths at these two-photon detun-ings. As Ω p = 0 . δ = − π ×
85 kHz. As Ω p = 0 . δ = − π ×
180 kHz. Both values of δ were determinedin the way described in Sec. V and shown by Fig. S4. InFig. S6, the gray dashed line is the beam width measuredwithout the presence of the atoms, and the black and redsolid lines are the linear best fits of the experimental data.It is verified by the gentle slopes of the best fits that theuncertainty in δ and the lensing effect played insignifi-cant roles in the measurement of the probe beam width.Thus, the reduction of the probe beam width is the con-sequence of the transverse momentum distribution of theRydberg polaritons being narrowed the thermalizationprocess. [1] B. Kim, K.-T. Chen, C.-Y. Hsu, S.-S. Hsiao, Y.-C.Tseng, C.-Y. Lee, S.-L. Liang, Y.-H. Lai, J. Ruseckas, G.Juzeli¯unas, and I. A. Yu, Effect of laser-frequency fluctu-ation on the decay rate of Rydberg coherence , Phys. Rev.A , 013815 (2019).[2] Y.-H. Chen, M.-J. Lee, I-C. Wang, S. Du, Y.-F. Chen,Y.-C. Chen, and I. A. Yu,
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