Nonlinear absorption in interacting Rydberg electromagnetically-induced-transparency spectra on two-photon resonance
Annika Tebben, Clément Hainaut, Andre Salzinger, Sebastian Geier, Titus Franz, Thomas Pohl, Martin Gärttner, Gerhard Zürn, Matthias Weidemüller
NNonlinear absorption in interacting Rydberg EIT spectra on two-photon resonance
Annika Tebben, Clément Hainaut, Andre Salzinger, Sebastian Geier, Titus Franz, Thomas Pohl, Martin Gärttner,
3, 1, 4
Gerhard Zürn, and Matthias Weidemüller Physikalisches Institut, Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany Center for Complex Quantum Systems, Department of Physics and Astronomy,Aarhus University, DK-8000 Aarhus C, Denmark Kirchhoff-Institut für Physik, Universität Heidelberg,Im Neuenheimer Feld 227, 69120 Heidelberg, Germany Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany (Dated: 24/02/2021 01:46:19 File: nonlinear_absorption)We experimentally investigate the nonlinear transmission spectrum of coherent light fields prop-agating through a Rydberg-EIT medium with strong atomic interactions. In contrast to previousinvestigations, which have largely focused on resonant control fields, we explore here the full two-dimensional spectral response of the Rydberg gas. Our measurements confirm previously observedspectral features for a vanishing control-field detuning that are explainable by existing theories, butalso reveal significant differences on two-photon resonance. In particular, we find qualitative defi-ciencies of mean-field models and rate-equation simulations in describing the nonlinear probe-fieldresponse under EIT conditions, suggesting spectral signatures of an interaction-induced resonancewith laser-dressed entangled pair states. While this effect is captured by the third-order nonlinearsusceptibility that accounts for pair-wise interaction effects, the experiments show that many-bodyprocesses beyond such two-body effects play a significant role already at surprisingly low probe-fieldintensities. These results suggest that a more complete understanding of Rydberg-EIT and emergingphoton interactions requires to go beyond existing simplified models as well as few-photon theories.
I. INTRODUCTION
Rydberg electromagnetically induced transparency(EIT) is nowadays a widespread and reliable tool for thecreation of strong optical nonlinearities based on Ry-dberg blockade induced dissipation [1, 2]. Ultimatelyreaching the regime of quantum many-body nonlinearoptics, that requires strong interactions between a largenumber of photons, would allow one to study fascinatingstrongly correlated states of light as for example photoncrystals and quantum fluids of light [3–7].On one hand, increasing the interaction strength perphoton is one possible route towards this goal [8]. Theregime of quantum nonlinear optics, where the opticaldepth per blockade radius is much larger than one [2],has successfully been reached experimentally [9] and thegeneration of dissipative [10], attractive [3, 11, 12], re-pulsive [13] and spin-exchange like [14] interactions hasbeen demonstrated. Moreover, applications as for ex-ample single-photon transistors [15, 16] and gates [17]became experimentally feasible. Beyond this, a first ex-perimental study of quantum nonlinear effects with anincreased number of photons has been reported [18].On the other hand, understanding high intensity Ry-dberg EIT, where the number of photons is large, is theother precursor towards quantum many-body nonlinearoptics with photons. This regime, where light fields canbe treated classically, is where the experimental inves-tigation of Rydberg EIT was initiated by the demon-stration of strong nonlinearities in a Rydberg gas underEIT conditions [19]. Thereafter, the role of interactionsin Rydberg EIT has been studied in detail [20–23] withmeasurements where either the probe or the control field resonantly couples its atomic transition and various the-oretical models have been put forward to describe theexperimental observations. While the developed theoryhas been successful in reproducing the general effects ofRydberg-state interactions on EIT in such systems, im-portant spectral details such as the appearance and ori-gins of nonlinear shifts and asymmetries are still underdebate [21, 22].Here, we address this question by a systematic studyof the spectral properties of EIT transmission in a coldatomic gas with strong Rydberg-state interactions bybroadly scanning both control- and probe-field frequen-cies. In particular, we measure the nonlinear absorptionspectrum of the probe-field on two-photon resonance, i.e.by simultaneously adjusting the control-field frequencyto maintain EIT conditions. As this implies a vanish-ing linear absorption for all probe-field frequencies, suchmeasurements directly probe pure interaction effects. Inturn, the measured spectra reveal qualitative discrep-ancies with existing theories that have previously gonemissing in experiments with resonant control fields [21–23]. Existing mean-field approximations and semiclas-sical rate-equation simulations both fail qualitatively toreproduce the observed broad absorption features andresonant nonlinear absorption away from single-photonresonance. While the latter can be qualitatively ex-plained within a low-intensity expansion [24, 25] thataccounts for resonances with control-laser dressed entan-gled pair-states [25–28], clear quantitative discrepanciessuggest that an improved understanding of Rydberg-EITrequires the inclusion of many-body effects beyond pair-wise atomic correlations.This article is organized as follows. Following a brief a r X i v : . [ phy s i c s . a t o m - ph ] F e b Figure 1. (a) Realization of a ladder-type Rydberg EIT sys-tem with counter-propagating probe (red) and control (blue)beams. Indicated detunings in the atomic level scheme arethe single-photon detuning ∆ = ∆ p and two-photon detuning δ = ∆ p + ∆ c , that are given by the laser detunings ∆ p,c ofthe probe and control beams, respectively. Rubidium Ryd-berg atoms interact via an interaction V ( r ij ) , which dependson the distance r ij between two atoms i and j . (b) Imaginarypart of the linear optical response | Im ( χ (1) ) | as a functionof the laser detunings. Transmission spectra are measured,where the control beam detuning ∆ c = 0 (green dashed line),or where only the single-photon detuning ∆ is changed whilestaying on two-photon resonance ( δ = 0 , red line). Here, theoptical response vanishes in the non-interacting regime. description of the experimental setup in Sec. II, wepresent the main results of the nonlinear transmissionmeasurements in Sec. III. In Sec. IV, we compare theobservations with the prediction of a mean-field approx-imation [22, 23, 29], Monte Carlo rate-equation simula-tions [30–33] and a theory based on the third-order sus-ceptibility that accounts for pair-wise atomic correlations[24, 25]. II. THE RYDBERG EIT SETUP
Our Rydberg EIT medium consists of a cigar-shaped Rb atom cloud with × × µ m 1 /e -waistsheld within an optical dipole trap, with a maximalpeak density of × cm − and with a tempera-ture of about µ K . As detailed in App. A we en-sure an accurate preparation of the ground state | g (cid:105) = | S / , F = 2 , m F = 2 (cid:105) , that together with the short-lived intermediate state | e (cid:105) = | P / , F = 3 , m F = 3 (cid:105) with decay rate γ e / π = 6 . MHz and a metastableRydberg state | r (cid:105) = | S / , m j = 1 / (cid:105) forms the three-level ladder EIT system, as depicted in Fig. 1(a). Ryd- berg atoms interact via isotropic van-der-Waals interac-tions V ( r ij ) = C /r ij , where C is the van-der-Waalscoefficient and r ij the distance between two atoms i and j . Counter propagating probe (
780 nm ) and control(
480 nm ) beams with Rabi frequencies Ω p and Ω c , couplethe ground to intermediate and intermediate to Rydbergstate transitions, respectively, and have linewidths be-low
530 kHz (see App. B and Sec. IV.D). The opticaldepth per blockade radius OD b is much smaller than onein our system, meaning that the probe field preserves itscoherent nature and can be treated classically [2].Earlier investigations of EIT spectra revealed, that theEIT area can act as a dispersive gradient-index lens [34].This is a consequence of a refractive index gradient, thatis induced by the inhomogeneity of a focused controlbeam within a uniform probe beam. In order to avoidthis effect and to ensure negligible dispersion, we invertthis geometry and choose a control beam with a waisttwice as large as the focused probe beam ( /e -beamwaist: µ m ).After releasing the atoms from the dipole trap, probeand control beam pulses are applied in order to createEIT conditions. We experimentally checked, that theused probe power and pulse times of to µ s are smallenough to avoid an avalanche creation of Rydberg ions[35]. Moreover, in order to adiabatically transfer theatoms into the EIT dark-state [36], we switch on the con-trol beam about µ s before we turn on the probe beampulse. This is in combination with a rise time of theprobe beam pulse of
50 ns , given by the AOM rise time,sufficient for an adiabatic preparation of the EIT-darkstate.After the EIT sequence we image the transmittedprobe light onto a CCD camera, using a 4f-imaging sys-tem with a resolution of approx. µ m (Rayleigh crite-rion). The pixel size of the CCD camera in the imagingplane is . µ m , which is small compared to the waist ofthe two laser beams. This allows to average the signal ofup to x pixel, corresponding to a maximal variation of in the probe beam Rabi frequency, in order to obtaina better signal-to-noise ratio for the determination of theprobe beam transmission T . The latter is defined by theratio of the measured transmitted light in the presenceand absence of the atomic cloud. III. TRANSMISSION MEASUREMENTS
In order to benchmark our Rydberg EIT systemagainst existing measurements [19, 21–23], we first recordthe probe beam transmission as a function of the probebeam detuning ∆ p while staying on resonance with thecontrol laser ( ∆ c = 0 ). This measurement followsthe green dashed line depicted in Fig. 1(b) and resultsin what is typically called an Autler-Townes spectrum.Afterwards, we present measurements on two-photonresonance by experimentally following the red line inFig. 1(b). Here, a new dimension of investigating Ryd- - -
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Figure 2. Autler-Townes spectra in the non-interacting (a)and interacting (b) regime. The measured probe beam trans-mission T against the probe detuning ∆ p is shown (blackpoints) for Ω p / Ω c = 0 . (a) and Ω p / Ω c = 0 . (b) for ∆ c = 0 and at a peak atomic density of (0 . ± . µ m − .The transmission spectra calculated with a mean-field model(blue solid line) and a MCRE simulation (orange dashed line)are depicted. Shaded areas take into account the uncertaintyin the atomic density. Ω c / π = 15 MHz and γ ge / π = 1 MHz for both theory curves. Gray arrows in (b) indicate the min-ima positions of the curve in (a). berg EIT is pursued, as the linear response of the mediumvanishes in the non-interacting regime. A. Non-interacting and interacting Autler-Townesspectra
When changing the probe beam detuning ∆ p withthe control beam on resonance ( ∆ c = 0 ) in the non-interacting regime, where the probe Rabi frequency issmall, we recover the known Autler-Townes spectrum, asshown in Fig. 2(a). A transmission of nearly at zero de-tuning and a symmetric spectrum supports the absenceof dephasing γ gr on the Rydberg coherence and thereforea largely coherent dynamics of a three-level system.In the interacting regime at a high probe Rabi fre-quency, as presented in Fig. 2(b), the transmission atzero detuning is reduced. Moreover, we observe a smallshift of the left minimum and an asymmetry of the spec-trum. These general features have also been observed inprevious measurements [21–23]. Solid and dashed linesin Fig. 2 are a the result of theoretical models and willbe discussed in Sec. IV. - -
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Figure 3. Transmission T on two-photon resonance ( δ = 0 ).(a) Measured transmission spectra for different ratios Ω p / Ω c .The gray dashed line indicates the measured transmissionin the absence of the control beam at ∆ = 0 . Other pa-rameters are Ω c / π = (28 . ± .
4) MHz and a peak atomicdensity ρ = (0 . ± . µ m − for the yellow curve and Ω c / π = (24 . ± .
2) MHz and ρ = (0 . ± . µ m − for all other curves. (b) Comparison of the measured trans-mission spectrum at Ω p / Ω c = 0 . , with the results of themean-field model (blue solid line), the MCRE simulation (or-ange dashed line) and the low-intensity theory (green dashed-dotted line). Shaded areas account for a variation of the den-sity of ± . B. Measurements on two-photon resonance
In order to investigate the effect of interactions on thetransmission spectrum in a different approach we performmeasurements on two-photon resonance by choosing ex-perimental parameters to follow the red line in Fig. 1(b),where the linear response of the medium vanishes in thenon-interacting regime. Precisely, we change the single-photon detuning ∆ while staying on two-photon reso-nance ( δ = 0 ) and record the probe beam transmissionas presented in Fig. 3(a).At low Ω p the probability to be in the Rydberg state issmall, such that in this so-called non-interacting regimenonlinearities due to Rydberg interactions are negligible.Here, the transmission is consistent with unity for allsingle-photon detunings, as shown by the yellow squaresin Fig. 3(a). This is expected as on two-photon resonancethe linear response of the atomic medium vanishes andEIT conditions are fulfilled.Gradually increasing the probe Rabi frequency in-creases the Rydberg state fraction, such that interactioneffects influence the dynamics. In this interacting regime,already for a ratio of Ω p / Ω c ≈ . , depicted by the blackcircles in Fig. 3(a), a dip in the transmission to about . appears on the negative detuning side. However, this dipis absent on the positive detuning side. For the experi-mental parameters of this measurement we can excludethat this absorption feature results from the influence ofstationary Rydberg excitations or ions in the medium.In order to support this statement, we present a mea-surement of Rydberg excitations after the EIT sequenceand an estimation of an upper limit for the absorptionresulting from these in App. C.Increasing the probe Rabi frequency further (red downtriangles) increases the strength of the absorption dip,but does not change its position. Moreover, the fea-ture is getting broader, but remains clearly visible until Ω p / Ω c ≈ . . For the highest ratio Ω p / Ω c ≈ . of thetwo Rabi frequencies (purple up triangles) strong absorp-tion continues to persists predominantly on the negativedetuning side but is shifted towards the single-photon res-onance and is further broadened. Therefore, the observedabsorption feature turns out to be very sensitivity with Ω p , which is a characteristic of a nonlinear phenomenon. IV. THEORETICAL MODELS
Having observed a small shift and an asymmetry in theinteracting Autler-Twones spectrum and a strong nonlin-ear absorption in spectra on two-photon resonance, wenow aim for a comparison of our measurements with ex-isting theoretical models. Working in the regime, wherethe probe field can be treated classically (OD b (cid:28) ),photon-photon and atom-photon correlations can be ne-glected [2]. However, interactions between atoms induceatom-atom correlations, that on one hand enable strongnonlinear effects and therefore make the system poten-tially useful for quantum optics applications, but on theother hand make a theoretical treatment of the RydbergEIT medium challenging. Exactly and numerically onlysolvable for a few atoms [19], methods to truncate thesecorrelations in the resulting many-body master equations[21, 30] need to be applied.The simplest truncation neglects direct two-body cor-relations in a mean-field approach and implements theresulting interaction induced shift and dephasing intothe single-body master equation [22, 23, 37]. The sameansatz of an interaction induced shift is followed in a rateequation model [30–33], that however solves the many-body rate equations using a Monte-Carlo simulation. An-other approach for truncation of the many-body correla-tions is a low intensity approximation of the optical Blochequations, in which two-body interactions can be treatedexactly [24, 25]. In the following we give some relevantdetails on these three approaches and comment on theirrange of validity. A. Mean-field model
Among other implementations of the mean-field model,we choose to compare our experimental results with an ansatz followed in [22], as experimental parameters, suchas the ratio Ω p / Ω c and the atomic density, are similar.In that work, the transmission of the probe field followsfrom the one-dimensional Maxwell Bloch equation, wherethe optical response of the medium enters in terms of amodel susceptibility ¯ χ = αχ B + (1 − α ) χ E [22]. Thismodel susceptibility is based on the solution for the sus-ceptibility χ of the single-atom master equation for anon-interacting three-level system and includes interac-tions in terms of level shifts. Two different parts χ B and χ E of the model susceptibility are weighted according tothe fraction α of all blockaded atoms excluding Rydbergatoms [22].Thereby, χ B describes the optical response of block-aded atoms and is given by a spatial integration of χ (∆ (cid:48) c = ∆ c + C /r ) over the radius r inside the block-ade radius. Here, interactions are induced as a level shiftand χ B equals the two-level susceptibility for strong in-teractions. χ E = χ (∆ (cid:48) c = ∆ c − ∆ R , γ gr = √ θ R ) ac-counts for interactions of unblockaded atoms with Ryd-berg excitations at a distance larger than the blockaderadius by introducing an average shift ∆ R and its vari-ance θ R , where the latter leads to a dephasing γ gr of theRydberg coherence [22]. In this model ∆ R and θ R canonly be approximated and are based on a local densityapproximation and an approximation for the Rydbergexcited fraction, that is derived from a semi-analyticalmodel using superatoms [38]. B. Monte-Carlo rate equation model (MCRE)
In this approach, the single-atom master equationwithout interactions is cast into a set of rate equa-tions by adiabatically eliminating the coherences [30–32]. Interactions are included as effective level shifts ∆ ( i ) int = (cid:80) j (cid:54) = i C /r ij for the Rydberg level of the i -thatom with distance r ij to atom j . Using a Monte-Carlosimulation, the many-body problem is solved by propa-gating the global ground state to the global steady state.In this Monte-Carlo rate equation model (MCRE) thepropagation of the probe field can be included by takinginto account the local probe Rabi frequency Ω ( i ) p , thatatom i experiences, for the calculation of the steady stateof atom i in each Monte-Carlo step [33]. For this pur-pose, the probe Rabi frequency is propagated throughthe cloud of randomly positioned atoms according to theone-dimensional Maxwell Bloch equation until atom i isreached. Thereby in each propagation step, the local at-tenuation experienced by the individual atoms that arepassed is subsequently accounted for. As a result, notonly global atomic observables, but also the probe beamtransmission can be simulated with this approach. C. Low-intensity theory
A more rigorous description of Rydberg-EIT can beobtained by expanding the many-body problem of theinteracting ensemble in terms of the number of Rydbergexcitations per blockade volume [2, 24, 25, 27]. Startingwith the underlying Heisenberg equations that describethe driven dynamics of the atomic states one obtains a hi-erarchy of equations for operator products that describecorrelations and entanglement induced by the strong Ry-dberg state interactions. Assuming that the Rydbergpopulation per blockade radius is small, this hierarchycan be truncated by neglecting three-body contributions,i.e. assuming that the probability to excite three nearbystrongly interacting Rydberg atoms is negligibly small.This permits to find an exact analytic solution for thethird order nonlinear susceptibility that fully accountsfor two-body correlations and entanglement on the levelof atomic pairs [25]. The transmission of the probe beamis then calculated by applying a local density approxima-tion and assuming spatially constant probe and controlbeams.
D. Range of validity
For a comparison of the range of the validity of thetheoretical models the strength of the applied fields, theatomic density as well as the interaction strength are con-sidered in the following.The mean-field model includes Rydberg interactionssolely as an interaction-induced energy shift based on theassumption that inter-atomic correlations can be com-pletely neglected. This requires that the mean distancebetween Rydberg excitations is larger than the blockaderadius, which is for example the case if Ω p / Ω c (cid:28) or fora small interaction strength. A simple mean-field modelhas been shown to fail to explain observations in coher-ent population trapping experiments as soon as excita-tion blockade becomes relevant [20], which depending onthe experimental parameters might already be the caseat low atomic densities. The mean-field model consid-ered here agreed well with an interaction induced shiftand dephasing observed in interacting EIT transmissionspectra for densities up to about . µ m − [22].The MCRE model also includes Rydberg interactionsas effective energy shifts but does not rely on assump-tions for calculating the average shift felt by one atom.Instead, it naturally includes the mean-field shift in a self-consistent manner and calculates the steady-state of the N -body density matrix. While still requiring Ω p / Ω c (cid:28) or Ω p / Ω c (cid:29) for atomic coherences to vanish [38], thesetwo aspects increase its range of validity to a large rangeof atomic densities and yields correct results also for den-sities as high as . µ m − [21, 31]. For Ω p / Ω c (cid:28) therate equation model was shown to agree well with the re-sult of a master equation calculation of four fully block-aded atoms independent of the driving strength Ω c /γ e [38]. As a result, the MCRE model was able to explaincertain aspects of interacting EIT spectra and the densitydependence of nonlinear absorption [21, 33].The low-intensity theory is based on a perturba-tive expansion in the probe field and therefore requires Ω p / Ω c (cid:28) . Its applicability in terms of atomic densitiesand interaction strength is combined in the requirement,that the Rydberg population per blockade volume needsto be much smaller than one [2]. In its form consideredhere, the low-intensity theory predicted the existence ofan enhanced nonlinear optical response for ∆ ∼ ± Ω c / as a consequence of a two-body, two-photon resonancein the non-adiabatic regime [25], but has not been com-pared to experiments in this regime yet. However, in theregime of large probe beam detunings, where the inter-mediate state can be adiabatically eliminated, the low-intensity theory was successfully compared to absorptionmeasurements showing the quadratic dependence on theprobe Rabi frequency at moderate densities [24]. V. COMPARISON BETWEEN THEORY ANDEXPERIMENT
We implement the three models [22, 25, 33] with atransversely constant probe beam intensity and a con-stant control beam intensity in all spatial dimensions.All three models account for the angle between thepropagation direction of the lasers and the main axis ofthe atomic cloud, as depicted in Fig. 1(a), and includethe Gaussian density distribution in propagation direc-tion. In the MCRE model the Gaussian density distri-bution in transversal direction is considered, while it isassumed to be constant for the other two models. Thisapproximation is justified due to the much larger waistof the atomic cloud in transversal direction compared tothe waist in propagation direction. We checked that inthe absence of interactions all three models coincide witheach other. A. Comparison with Autler-Townes measurements
The Rydberg population per blockade volume of theinteracing Autler-Townes measurement is . on reso-nance and increases off-resonance even further, such thatit cannot be considered much smaller than one. There-fore, we omit a comparison of the low-intensity theorywith the Autler-Townes measurement in the following.The ratio of the two Rabi frequencies as well as theatomic density are in a regime where a comparison withthe other two models is possible.In order to match the result of the mean-field modeland the MCRE simulation to our measured transmissionspectra in Fig. 2, we use for both models a dephasing γ ge / π = 1 MHz of the excited state coherence to ac-count for both the laser linewidth of the probe beamand a density dependent dephasing present in the system(see App. B). Dephasing due to laser noise was indepen-dently determined in a measurement of the two-photonlinewidth and found to be well below
10 kHz . As this issmall compared to the Rydberg decay rate of . ,we set γ gr = 0 in the following. Moreover, the probeRabi frequency is varied simultaneously for both theorieswithin its systematic error of .In the non-interacting regime the mean-field model(blue solid line) and the result of the MCRE simula-tion (orange-dashed line) agree well with the measuredtransmission spectrum in Fig. 2(a). The slight deviationsobserved can come from a small fluctuation of the con-trol beam power and a possible small misalignment ofthe counter-propagating beams, which are not includedin the theoretical models.In the interacting regime, shown in Fig. 2(b), a reduc-tion of the transmission around ∆ p = 0 and an asym-metry in the spectrum is predicted by the two models.However, while both theories predict a shift of the res-onance position to ∆ p / π ≈ , we do not observethis large shift in the experiment. The deviation of thetransmission predicted by the two theories around single-photon resonance can be explained by the different im-plementation of the interaction-induced level shift andits variance in the two models (see App. D for details).Moreover, the observed lower transmission on resonancecan be captured by including an effective dephasing rate γ gr into the models, which could be explained by Ry-dberg excitations that might be present in the mediumfor the experimental parameters of this measurement asdiscussed in App. C.While the attenuation of the transmission on resonanceis a consequence of Rydberg blockade induced absorptionand an experimentally and theoretically approved featureof interacting Rydberg EIT systems [2], the absence orpresence of a shift and asymmetry in the spectrum is de-bated in literature [21, 22]. In theories, that rely on amean-field shift of the Rydberg level, as the consideredmean-field and MCRE models, the asymmetry and shiftare a consequence of an anti-blockade effect. It allows theexcitation of Rydberg pair states for a positive probe de-tuning, thereby reduces absorption and effectively shiftsthe resonance position [21]. This shift is also observablein the solution of the master equation for a few atoms[33].In the first experimental demonstration of nonlineari-ties in a Rydberg EIT medium, no shift and asymmetrywas measured [19] in the EIT spectrum. This fact was ex-plicitly attributed to the absence of Rydberg excitationsor ions in the system, which could cause a mean-fieldshift, and was explained as a sole cooperative nonlinear-ity. Subsequent publications showed measurements thatexhibited both a shift and asymmetry in the EIT as wellas the Autler-Townes regime [21–23], but also discussedthe absence of the asymmetry as the result of increasedabsorption due to interaction induced motion in the timeof one experimental cycle [21]. B. Comparison with measurements on two-photonresonance
For the measurements on two-photon resonance theatomic density is in a regime where all three modelsshould be applicable. Moreover, the Rydberg popula-tion per blockade volume is below . for the yellowand black curves in Fig. 3(a) for all detunings, but ex-ceeds this threshold for the other two curves. This meansthat at least for the yellow and black curves, for which Ω p / Ω c (cid:28) , the requirements for all three models arefulfilled.In the non-interacting regime on two-photon resonance( δ = 0 ), the transmission is nearly for all single-photondetunings ∆ , as shown by the yellow squares in Fig. 3(b).This shows that experimental imperfections, which wouldlead to single-particle dephasing (e.g. atomic motion,imperfect initial state preparation, remnant DC electricfields), are negligible. In theoretical models a transmis-sion of is expected, as on two-photon resonance thepopulation in the Rydberg state and thus interaction in-duced shifts tend to zero for small Ω p or small atomicdensities. In combination with χ ( δ = 0) = 0 for negli-gible single-particle dephasing this results in a vanishinglinear response. All three models reproduce this behaviorcorrectly.For the interacting regime, Fig. 3(b) shows a compari-son of the measured transmission spectrum for Ω p / Ω c =0 . with the three different models. For all of them theexperimental parameters are directly used as an input forthe models and the dephasings γ ge and γ gr are the sameas for the Autler-Townes measurements. Apparently allthree models fail to describe our measurement.As shown in Fig. 3(b), only qualitatively one absorp-tion dip on the negative detuning side is found with themean-field and MCRE models. However, its position de-viates from and cannot be superimposed with the mea-sured one by changing parameters, as for example theatomic density, within an acceptable range with respectto the experimental parameters. The stronger absorp-tion predicted by the mean-field model compared to theMCRE simulation stems from the inclusion of the vari-ance θ R of the interaction induced shift in the mean-fieldmodel, which becomes more important as the fraction α of all blockaded atoms excluding Rydberg atoms issmaller than . for all detunings. This variance is notexplicitly included in the MCRE simulation (see App. Dfor details).The low-intensity theory predicts two transmissionminima as a consequence of two-body two-photon res-onance. However, even though the assumptions for thismodel are met, it can not explain the absorption featureon the negative detuning side. C. Discussion
Similar to previous experiments [19, 21–23], the dis-cussed theoretical calculations capture the nonlinear be-havior of the measured Autler-Townes spectra for res-onant control-laser fields besides the absence of a shiftthat has already been debated [21, 22]. Strikingly, how-ever, all three approaches fail to explain the observednonlinear absorption spectrum under conditions of EIT.The outlined mean-field description [22] as well as theMCRE simulations [33] include interaction effects in anapproximate way that augments the single-atom equa-tion of motion by a collective level shift produced by sur-rounding atoms. Consequently, these approaches do notfully account for correlations and entanglement betweenatoms that arises from strong pair-wise interactions inthe presence of laser driving. In particular, they neglectpair-state resonances that emerge from strong control-field coupling of blockaded atom pairs [25–28] and leadto enhanced nonlinear absorption around ∆ ∼ ± Ω c / .While the presented low-intensity theory [25] exactly ac-counts for this effect on a two-body level, the comparisonto our experiments suggests that the collective influenceof multiple interacting atoms plays a significant role forthe observed nonlinear absorption spectrum.While an exact description of the laser-driven interact-ing Rydberg atom ensemble is numerically intractable,future improvements of current theoretical approachesmay shed light on the discrepancies revealed in this work.For example, a hybrid MCRE scheme, proposed in [32],combines the described rate equation description of manyinteracting atoms with an exact treatment of two-bodyquantum dynamics. Hereby one identifies close lyingatoms that form otherwise isolated pairs, for which thecorresponding two-body master equation is solved ex-actly to obtain a corresponding system of two-body rateequations that yields the exact two-body steady state.This approach is expected to provide an improved de-scription of the nonlinear absorption at low atomic den-sities [32]. However, the identification of atomic pairsbecomes ambiguous at our densities and would, hence, re-quire a quantum master-equation description of extendedatomic clusters in future theoretical work. Moreover, theoutlined low-intensity theory for the third-order opticalsusceptibility of the Rydberg gas [25] may be expandedby truncating the underlying hierarchy of operator equa-tions, discussed in Sec. IV C, via a closure relation thattakes into account the effect of multiple surrounding Ryd-berg atoms beyond direct two-body terms. For example,this could be achieved within a systematic cluster expan-sion and ladder approximation of three-body terms toinclude the mean-field level-shift generated by Rydbergatoms surrounding a given atomic pair. VI. CONCLUSION AND OUTLOOK
In conclusion, we have experimentally investigatedthe nonlinear absorption spectrum of a Rydberg-EITmedium with strong atomic interactions. Our measure-ments of the nonlinear behavior of the Autler-Townesabsorption peaks for resonant control-fields confirm pre-viously observed spectral features that are explainableby existing theories. Strikingly, however, we found sig-nificant deviations for the probe-field absorption spec-trum on two-photon resonance, i.e. when simultaneouslyscanning both laser frequencies to maintain EIT condi-tions. The appearance of such discrepancies that implyqualitative deficiency of existing theories comes as a sur-prise, in light of the substantial previous investigations ofRydberg-EIT in the semi-classical [19–29, 33, 34, 38] aswell as quantum regime [2, 3, 9–18]. Maintaining linearEIT, implies that any absorption predominantly arisesfrom nonlinear effects such that the presented measure-ments provide a more stringent test to the theoreticalunderstanding of the underlying optical nonlinearities.The detailed comparison to different and complementarytheoretical approaches, presented in this work, indeedsuggests that an improved treatment of the driven many-body dynamics is necessary to describe EIT in interactingatomic gases.
ACKNOWLEDGMENTS
The authors gratefully acknowledge insightful discus-sions with Valentin Walther and Yong-Chang Zhang.This work is part of and supported by the DFG Pri-ority Program "GiRyd 1929" (DFG WE2661/12-1), theDNRF through the Center for Complex Quantum Sys-tems (Grant agreement no.: DNRF156), the CarlsbergFoundation through the Semper Ardens Research ProjectQCooL, the EU through the H2020-FETOPEN GrantNo. 800942640378 (ErBeStA), and the Heidelberg Cen-ter for Quantum Dynamics, and funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation, Project-ID 273811115, SFB 1225). A.T. acknowl-edges support from the Heidelberg Graduate School forFundamental Physics (HGSFP). C.H. acknowledges sup-port from the Alexander von Humboldt foundation.
Appendix A: Preparation of the three-level system Rb atoms are loaded into a magneto-optical trap(MOT) from a high flux cold atom source [39]. Af-ter a compressed and a dark-MOT phase [40, 41] theatoms are transferred into a far detuned, crossed opti-cal dipole trap. Using a combination of optical pumpingand Landau Zener transfers between hyperfine sublevelsthe atoms are prepared in the hyperfine ground state | g (cid:105) = | S / , F = 2 , m F = 2 (cid:105) in the presence of a
30 G magnetic field. With this procedure we achieve a cigar - - -
10 0 10 20 300.02.04.0
Figure 4. Optical depth OD = − ln ( T ) as the function of theprobe beam detuning ∆ p in the absence of the control beam( Ω c = 0 ) and for peak atomic densities of ρ ≈ . µ m − (a) and ρ ≈ . µ m − (b), respectively. The result of themean-field model, with the dephasing γ ge of the excited statecoherence as the only free fitting parameter is shown as asolid line. The shaded area indicates a variation of the atomicdensity by ± . shaped atomic cloud with × × µ m 1 /e -waists,a maximal peak density of × cm − and a temper-ature of about µ K in a well define initial state | g (cid:105) .The probe and control beams that couple the groundto the Rydberg state in a two-photon process via the in-termediate state are right- and left-circular polarized, re-spectively, with respect to the applied magnetic field. Incombination with the careful preparation of the groundstate this ensures that our EIT setup is realized as a welldefined three-level-ladder system. Moreover, both lasersare locked to a stable, high finesse Fabry-Pérot cavitywith a free spectral range of . , allowing for detun-ings ∆ p,c of the two beams of up to
750 MHz and smalllaser linewidths below
10 kHz . Appendix B: Characterization of dephasing in thetwo-level system
In order to determine the dephasing γ ge of the excitedstate coherence, we measure the optical depth OD of theatomic cloud as a function of the probe beam detuning ∆ p in the absence of the control beam ( Ω c = 0 ). Forcomparability with our measurements in the main textwe choose a rather high probe Rabi frequency Ω p / π =1 . .In the low density regime, shown in Fig. 4(a), we ex-tract a dephasing of γ ge / π = (0 . ± .
05) MHz us-ing the mean-field model. Here, the error accountsfor shot-to-shot fluctuations of the atomic density by ± . In the high density regime, as depicted inFig. 4(b), we obtain a rather large dephasing of γ ge / π =(1 . ± .
1) MHz . - -
10 0 10 200102030
Figure 5. Measurements of ion counts on two-photon reso-nance ( δ = 0 ) for different ratios Ω p / Ω c . The ions wheredetected simultaneously to the measurement of Fig. 3. The natural linewidth due to population decay, powerbroadening as well as a reduction of the linewidth dueto propagation effects are intrinsically included into themean-field model, such that these effects can not be thesource of the observed dephasing. Density-dependent de-phasing mechanism that could cause such a broadeningare atomic collisions or rescattering of photons. Esti-mating the broadening due to collisions [42] by calculat-ing the collision rate from the atomic velocity and themean-free-path shows, that the temperature or the den-sity of the atomic gas are too low for explaining this largeamount of dephasing. However, due to the large extentof the atomic cloud transversal to the propagation direc-tion, the transverse optical depth is large and allows formultiple rescattering of the photons [43, 44], which canbroaden the line at high densities.
Appendix C: Rydberg excitation measurement ontwo-photon resonance
Besides a transmission measurement, our setup alsoallows to detect Rydberg excitations, that remain in theatomic cloud after turning off the EIT lasers, by fieldionization of the atom cloud and subsequent detection ofthe resulting ions on a micro-channel plate (MCP). Here,the detection efficiency, measured by depletion imaging[45], is about . ions per Rydberg excitation and inthe absence of the control beam, where no ions can becreated, we measure . ± . counts, where the erroris the standard error of the mean, setting a threshold forthe detection of Rydberg excitations.For the measurement on two-photon resonance pre-sented in Fig. 3(a) we simultaneously recorded the ioncounts on the MCP, as shown in Fig. 5. For the ratioof Ω p / Ω c = 0 . , where the nonlinear absorption in thetransmission measurements appears, the number of de-tected ions is about one for all detunings, as shown bythe black circles in Fig. 5. When increasing the ratio ofthe two Rabi frequencies further, the ion count increasesto about counts, but stays approximately constant overthe whole range of single photon detunings. For the high-est measured ratio the number of detected ions increasessignificantly with a maximum around zero single-photondetuning.In the case of a coherent evolution in the EIT sys-tem we would expect to detect no ions after the EITsequence. Therefore, we attribute the observed ions athigh ratios of the two Rabi frequencies to stationary Ry-dberg excitations in the medium. As these excitations donot get downpumped by the control beam into the de-caying intermediate state it is to be presumed that theseare excitations in other than the | S / , m j = 1 / (cid:105) Ry-dberg state. An explanation for the creation of thesesexcitations might be radiation trapping [43, 46] and sub-sequent state-changing collisions or anti-blockade excita-tion of Rydberg states that are not coupled by the controllaser. Theses unwanted Rydberg excitations have alreadybeen observed and termed "Rydberg pollutants" in [18].For the atomic density and the ratio of the two Rabifrequencies used for the interacting Autler-Townes mea-surement presented in Fig.2(b), which lay between theblack and red curves in Fig. 5, the ion measurement sug-gests the presence of a large number of Rydberg excita-tions that could lead to the observed slightly lower trans-mission around resonance.However, for the measurement at Ω p / Ω c ≈ . wemeasure approximately one ion count at the position ofthe transmission dip at ∆ min = − . According tothe detection efficiency, this sets an upper bound of onthe number N Ryd of Rydberg excitations in the medium.We now consider a worst case scenario in order to esti-mate the maximal influence of these excitations on theprobe beam propagation. For this purpose we assume,that all these excitations are located in the integratedregion of x pixels, where the probe beam transmissionis evaluated. Furthermore, we assume that these excita-tions are atoms in the P Rydberg state, that processstrong dipolar interactions with the S Rydberg statewith a coefficient c of about . µ m .We estimate the resulting absorption from theses ex-citations as follows: Each Rydberg excitation rendersthe medium absorptive in a spherical volume given bythe blockade radius R b , which is approximately . µ m at ∆ min . Assuming that all excitations are placed ina chain behind each other the resulting optical depthOD Ryd = OD off R b N Ryd /L can be calculated from thepropagation distance L through the whole atomic cloud,the peak atomic density ρ , which is given in the captionof Fig. 3, and the off-resonant optical depth OD off = σ off ρ L of two-level atoms. Here, the off-resonant crosssection σ off = aσ is the resonant cross-section σ mul-tiplied by a factor a = 0 . that takes into accountthe Lorentzian lineshape of the two-level absorption withdecay rate γ e . In a last step we have to account forthe fact, that the transversal size A Ryd = πR b of oneblockaded volume is smaller than the evaluated pixelarea A = (4 × . µ m) on the CCD camera and usethe scale s = A Ryd /A to finally obtain the transmission T Ryd = (1 − s ) + s exp( − OD Ryd ) ≈ . in the presence of ten Rydberg excitations.Overall, this estimation in a worst case scenario resultsin an upper bound of for the probe beam absorptionsolely due to these Rydberg excitations. Therefore, forthe ratio of Ω p / Ω c = 0 . , unwanted Rydberg excitationscan not explain the observed strong absorption feature. Appendix D: Comparison of mean-field and MCREmodel
In the spectra on two-photon resonance a stronger ab-sorption is predicted by the mean-field model than by theresult of the Monte-Carlo rate equation model, shown inFig. 3(b). We explain in the following that this resultsfrom the assumption of how the interaction-induced levelshift is included in the two models.On one hand in the MCRE model, the totalinteraction-induced level shift ∆ ( i ) int felt by an atom i isdetermined by the sum (cid:80) j (cid:54) = i ∆ ij = (cid:80) j (cid:54) = i C /r ij over allshifts induced by the surrounding Rydberg atoms [33].As the MCRE simulation is seeded with a distribution ofatoms according to the geometry of the experiment, theinter-atomic distances r ij vary, which immediately leadsto a certain variation of the level shifts ∆ ij .On the other hand, the considered mean-field model isbased on the non-interacting single-body susceptibilityand includes an interaction-induced level shift therein[22]. This means that nothing like an atomic distribu-tion, and therefore no variance of the level shift is con-sidered a priori. For distances smaller than the block-ade radius, the level shift is completely determined byan integration over the radius r and the resulting sus-ceptibility χ B inside the blockade radius is therefore un-ambiguously defined. However, for the susceptibility χ E outside the blockaded sphere assumptions about the av-erage level shift ∆ R and its variance θ R have to be made.Han et al. [22] calculate both based on a mean-field as-sumption as well as on an assumption for the Rydbergexcitation fraction. The variance is then included as aneffective dephasing γ gr = √ θ R of the Rydberg coherencein the single-body susceptibility [22]. Finally, the vari-ance, which is proportional to the Rydberg excitationfraction, determines the weight between the two partsof the overall model susceptibility ¯ χ = αχ B + (1 − α ) χ E [22]. This implies, that the more one enters the blockadedregime (large α ), the less weight is put on the assumptionmade for the variance of the level-shift entering χ E .For the Autler-Townes measurement in the interactingregime, α is larger than . around the Autler-Townestransmission minima. Hence, the contribution of χ B and χ E are quite similar, such that the relative importance ofincluding a variance of the level shift is small. As a result,the mean-field and MCRE model give similar predictionsfor the transmission spectrum. Only around resonance,where α is about . , deviations between the two modelsstart to appear.For the measurement on two photon resonance α < - -
10 0 10 200.00.51.0 - -
10 0 10 20 -- -
10 0 10 20 - -
10 0 10 200.80.91.0 - -
10 0 10 200.00.050.10
Figure 6. Comparison of the mean-field model (blue: θ R (cid:54) = 0 , purple: θ R = 0 ) and MCRE simulation (orange dashed lines) forthe parameters of Fig. 3(b). (a) Autler-Townes transmission spectra as a function of the probe beam detuning ∆ p for differentcontrol beam detunings ∆ c / π = { , , − } MHz. (b) Transmission T against the single-photon detuning ∆ on two-photonresonance ( ∆ c = − ∆ p ), as in Fig. 3(b). Black points depict the measured spectrum and the result the two models theorymodels are shown. Colored circles indicate the transmission values of the corresponding Autler-Townes spectrum in (a). (c) α of the mean-field model against the single-photon detuning ∆ . For all theory curves the parameters are the same as in Fig. 3(b).For a discussion of the curves see the main text. . for all single-photon detunings ∆ and is especiallyonly about . at ∆ / π = − , as shown inFig. 6(c). At the same detuning the effective dephasingon the Rydberg coherence √ θ R / π is as large as ,putting a lot of weight on the assumptions made in themean-field model.In order to show, that the transmission curve on two-photon resonance predicted by the mean-field model isdominated by the variance θ R , we show in Fig. 6(a) theAutler-Townes transmission spectra for three differentdetunings ∆ c / π = { , , − } MHz corresponding to thepoints marked in the curves on two-photon resonance de-picted in Fig. 6(b). The mean-field model with (without, θ R = 0 ) the variance θ R is shown in blue (purple) andthe result of the MCRE simulation is shown in orange forcomparison.For positive single-photon detunings ∆ > , where α is larger, the mean-field model and the MCREmodel almost agree for the Autler-Townes spectrum with ∆ c / π = − and setting θ R = 0 makes them almostidentical. 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