Electromagnetically Induced Transparency in strongly interacting Rydberg Gases
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Electromagnetically Induced Transparency in strongly interacting Rydberg Gases
C. Ates, S. Sevin¸cli, and T. Pohl
Max Planck Institute for the Physics of Complex Systems,N¨othnitzer Strasse 38, 01187 Dresden, Germany (Dated: September 24, 2018)We develop an efficient Monte-Carlo approach to describe the optical response of cold three-level atoms in the presence of EIT and strong atomic interactions. In particular, we consider a”Rydberg-EIT medium” where one involved level is subject to large shifts due to strong van derWaals interactions with surrounding Rydberg atoms. We find excellent agreement with much moreinvolved quantum calculations and demonstrate its applicability over a wide range of densities andinteraction strengths. The calculations show that the nonlinear absorption due to Rydberg-Rydbergatom interactions exhibits universal behavior.
The effect of electromagnetically induced transparency(EIT) [1] in light-driven multi-level systems continues toplay a pivotal role in quantum and nonlinear optics. En-abling slow light propagation and thus long photon inter-action times at low loss levels [2, 3], EIT media provide apromising route to applications in optical communicationand quantum information science. Optical nonlinearities,however, typically arise from higher order light-atom in-teractions, such that realizations of such applications atvery low light intensities [4] remain challenging. In thisrespect, recent experimental studies of EIT in cold Ry-dberg gases [5–9] are opening up new perspectives fornonlinear optics on a few photon level. Exploiting theexaggerated properties of Rydberg atoms and, in partic-ular, the strong interactions among the atoms, nonlinearphenomena can be greatly enhanced in ultracold Ryd-berg gases.Recently, several different methods have been used tostudy laser-driven interacting gases [10–14]. A simul-taneous treatment of EIT and long-range interactions,however, poses additional challenges. Common meanfieldapproaches, as successfully applied to radiation trappingeffects on EIT [10], are found to fail [11] due to the largestrength and enormous range of the van der Waals in-teraction, which lead to non-negligible correlations be-tween the atoms. On the other hand, exact descriptions,based on Hilbert space truncation by interaction-blockedmany-body states [12, 13], are also inapplicable sincethere always remains an exponentially large number ofmany-body states involved in the interaction-free probetransition.Here we present a theoretical approach that allows toobtain the fully correlated steady state populations viaclassical Monte Carlo sampling. This is shown to yieldthe nonlinear optical response to classical light fields inthe presence of arbitrarily strong atomic interactions. Acomparison to reduced density matrix calculations showsvery good agreement for small and moderate densities.Upon proper scaling the simulation results reveal a uni-versal behavior of the nonlinear absorption, which illus-trates the role of Rydberg atom interactions in the emer-gence of dissipative photon-photon interactions.The considered Rydberg-EIT level scheme is shown inFig.1. The signal laser couples the ground state | i to a | (cid:127) | (cid:127) | (cid:127) γ γ | (cid:127) | (cid:127) | (cid:127) C r ij atom i atom j r ij ∆ ∆ Ω Ω Ω Ω ∆ ∆ FIG. 1. Illustration of the considered three-level ladderscheme. Two laser fields successively couple the states | i , | i and | i . For isolated atoms, EIT is realized on two-photonresonance ∆ = − ∆ . The van der Waals interaction shift C /r ij between atoms in the Rydberg state | i modifies thisideal transparency and leads to a nonlinear optical responseof the medium. low lying state | i with Rabi frequency Ω . State | i iscoupled by a strong control laser (Ω > Ω ) to a highlyexcited Rydberg state | i . The resulting dynamics ofa gas composed of N such independent atoms is, thus,governed by the Hamiltonian H = − N X i =1 (∆ | i ih i | + (∆ + ∆ ) | i ih i |− Ω | i ih i | − Ω | i ih i | + h.c.) . (1)In addition, the intermediate state | i radiatively de-cays with a rate γ , whereas spontaneous decay of thelong-lived Rydberg level can safely be neglected. Forresonant driving, each atom settles into a dark state, | d i i ∼ Ω | i i − Ω | i i , which is immune to the laser cou-pling and radiative decay [1]. Consequently, the complexsusceptibility χ of the lower transition vanishes and themedium becomes transparent.In the presence of van der Waals interactions U = X i 60 this yields anenhancement of about 10 compared to interactions inlow-lying states, implying a drastically different excita-tion dynamics. On the one hand, the resulting level shiftslead to a strong suppression of Rydberg excitation[15],due to an excitation blockade of close atoms [16]. Onthe other hand, the interactions perturb the atomic darkstates [17, 18], and admix the dissipative intermediatestate [9, 11]. To account for these effects, we start fromthe von Neumann equation i ˙ ρ ( N ) = [ H + U ] − i L [ ρ ( N ) ] , (3)for the N -body density matrix ˆ ρ ( N ) of the gas. The Lind-blad operator L describes spontaneous decay of the in-termediate state, but can also include finite laser bandwidths, denoted by γ and γ , respectively. To obtain the steady state populations of eq.(3), wetransform it to a many-body rate equation. For clar-ity we start from the simple case C = 0 for which the N -particle density matrix factorizes, and eq.(3) reducesto a set of single-atom optical Bloch equations. Uponadiabatic elimination the dynamics of the atomic levelpopulations followsdd t ρ ( i )11 ρ ( i )22 ρ ( i )33 = − a a a a − a a a a − a ρ ( i )11 ρ ( i )22 ρ ( i )33 , (4)where the coefficients a αβ are straightforwardly obtainedas a function of the laser parameters.This is a common approximation to the long-time dy-namics of two-level systems [19]. Applications of the de-scribed elimination procedure to three-level atoms are,however, rather scarce, since it often leads to negativetransition rates [20]. To resolve this obstacle we proposea linear transformation that removes the negativity and,at the same time, preserves the correct steady states ofthe underlying von Neumann equation (3). For Ω < Ω this is accomplished by adding a correction matrix∆ a = σ (1 − R ) + σ σ (1 − R ) − σ + σ − σ R − σ (1 − R ) − σ R σ R + σ (1 − R ) σ − σ + σ R − σ + σ R − σ R − σ (1 − R ) − σ (1 − R ) , (5)to the original coefficient matrix a in eq.(4), where σ αβ =( | a αβ | − a αβ ) / R αβ are free parameters. With thisdefinition of the σ αβ any negative rate coefficient a αβ isset to zero, while the additional terms compensate for theaccording changes of the steady states. Consequently, theparameters R αβ are chosen such that the transformedrate equations ˙ ρ ( i ) = ( a + ∆ a ) ρ ( i ) yield steady statesidentical to those of the original equation ˙ ρ ( i ) = a ρ ( i ) .Explicitely, this condition gives R = a ( a + a ) + a ( a − a ) a ( a + a ) + a ( a + a ) (6a) R = a ( a + a ) + a ( a − a ) a ( a + a ) + a ( a + a ) (6b) R = a ( a + a ) + a ( a + a ) a ( a + a ) + a ( a − a ) (6c)for the transformation coefficients in eq.(5).Having established a proper rate equation descriptionfor the single particle dynamics, the approach can be ex-tended to interacting atoms. As shown in [21], this isstraightforwardly accomplished by replacing the upperdetuning of each atom by ∆ ( i )2 = ∆ − P ′ j = i U ij . Here,the sum only runs over atoms in the Rydberg state | i ,such that the local detunings and, hence, the individual atomic transition rates become dynamical variables thatdepend on the entire many-body state of the system. Asa result one obtains a many-body rate equation wherethe dynamics of the i th atom is governed by its indi-vidual transition rates ˜ a ( i ) αβ which depend on the actualRydberg atom configuration through the local detuning∆ ( i )2 . Although this rate equation still covers the expo-nentially large number of all 3 N many-body states, itcan be efficiently solved via classical Monte-Carlo sam-pling. Starting from the initial state with all atoms intheir ground state | α i = | i , the steady state is obtainedby performing repeated random transitions according tothe probabilities p ( i ) α → β = δt · ˜ a ( i ) αβ to make a transitionwithin a time step δt . In this way, we are able to ob-tain the fully correlated N -body state-distribution of theparticles, and in particular the average level populations ρ αα = N − P i ρ ( i ) αα . SinceIm( ρ ) = γ Ω ρ , (7)the Monte-Carlo approach also allows to determine theimaginary part of the complex optical susceptibility χ = 2 µ ρ ǫ ~ Ω ρ , (8) -3 -2 -1 0 1 2 30.030.060.090.120.15 ∆ [MHz] ρ = 3 · cm − ρ = 10 cm − ρ = 5 · cm − I m [ χ ] (cid:127) µ ρ ǫ (cid:127) Ω FIG. 2. Calculated probe beam absorption spectra for aRubidium Rydberg-EIT medium at various densities. Thelines show results of the Monte-Carlo simulations comparedto quantum calculations based on a reduced-density matrixexpansion [11] (symbols). The atoms are resonantly (∆ = 0)excited to 55 S Rydberg states. The probe and coupling beamshave Rabi frequencies of Ω = 1MHz and Ω = 2MHz, respec-tively, with linewidths of γ = γ = 100kHz. where µ denotes the dipole moment of the probe tran-sition and ǫ is the permittivity of vacuum. In addition,the resonant real part of χ can be obtained from thenonlinear absorption spectrum using the Kramers-Kronigrelations.In the following we consider the specific case of a Ru-bidium Rydberg gas. The atoms are excited by theprobe laser (Ω ) from the | S / i = | i ground stateto the | P / i = | i intermediate state, while the cou-pling laser resonantly (∆ = 0) drives the transitionbetween | P i and a | nS / i = | i Rydberg state withΩ = 2MHz. We include the intermediate state decay of γ = 6 . γ = γ = 100kHz for both beams.Fig.2 shows the calculated absorption spectrum at dif-ferent densities for Ω = 1MHz and n = 55. In order tocheck our results at low densities we have also performedsimulations based on a reduced-density expansion of thevon Neumann equation (3), as described in [11]. Theexcellent agreement between these entirely different cal-culations attests to the quality of both approaches. Atthe lowest density, Rydberg-Rydberg atom interactionsare ineffective, giving a small resonant absorption duethe finite laser linewidths γ and γ . At higher den-sities the interactions lead to a significant suppressionof the resonant transmission. In accord with recent ex-periments [9], the position and width of the absorptionminimum are, however, largely unaffected by the interac-tions, which are as large as C (4 πρ/ ≈ · cm − . Apparently, this is due tothe van der Waals blockade that prohibits simultaneousRydberg excitation of close atoms.The density dependence of the complex susceptibilityis shown in Fig.3. For low and moderate densities our Monte Carlo calculations and the reduced-density matrixexpansion [11] give consistent results for both the real andimaginary part of ρ (and hence of χ ). As expectedfrom a low-density expansion the latter tends to deviatewith increasing densities, with significant deviations oc-curring only at rather high densities & cm − . Aswe show in the following, the Monte-Carlo calculationsyield the expected high density limit, and should thusapplicable for arbitrary densities.As the density increases the imaginary part of χ starts to saturate while the real part develops a maxi-mum and goes to zero at high densities. In fact, thisbehavior can be understood from simple arguments. Atvery high densities, a single Rydberg atom strongly shiftsa large number of surrounding atoms out of resonance.As a consequence the Rydberg laser appears far detunedfor the majority of atoms, such that they act as an effec-tive two-level medium. Taking the limit ∆ ( i )2 → ∞ theasymptotic high-density steady state value of ρ , hence,approaches Im[ ρ ( ∞ )12 ] = Ω γg γ + g Ω (9)where g = ( γ + γ ) /γ , g = (2 γ + 3 γ ) / ( γ + γ ).This simple limit is indicated by the horizontal arrowin Fig.3 and gives very good agreement with the MonteCarlo simulations.It thus appears reasonable to employ the derived high-density limit as a natural scale for χ . In addition we canuse the interaction blockade of Rydberg excitation [16] torescale the gas density, i.e. the abscissa in Fig.3. Com-paring the Rydberg atom density ρ ryd = ρ N − P i ρ ( i )33 obtained from the Monte Carlo calculation to the corre-sponding value ρ (0)ryd for vanishing interactions one obtains ρ [cm − ] χ (cid:127) µ ρ ǫ (cid:127) Ω Im[ χ ] Re[ χ ] Im[ χ ( ∞ )12 ] FIG. 3. Complex susceptibility χ as a function of the gasdensity ρ under EIT conditions, ∆ = ∆ = 0. The reduceddensity matrix calculations (symbols) start to deviate fromthe Monte Carlo results (lines) for densities & cm − . Re-maining parameters are identical to those of Fig.2. -2 -1 ρ [cm − ] f bl χ (cid:127) µ ρ ǫ (cid:127) Ω ˜ χ (a) (b) FIG. 4. (a) Nonlinear absorption coefficient Im[ χ ] as a func-tion of the density of Rubidium atoms for Ω = 2MHz and γ = γ = 0. The different symbols correspond to differ-ent principal quantum numbers and probe Rabi frequencies:Ω = 1MHz, n = 50 (squares), Ω = 0 . n = 50 (cir-cles), Ω = 1MHz, n = 70 (triangles) and Ω = 1MHz, n = 50(diamonds). (b) After proper scaling [cf. eqs.(10) and (9)] alldata points follow the simple universal scaling relation eq.(11)(line). the fraction f bl = ρ (0)ryd ρ ryd − χ = χ /χ ( ∞ )12 on the laser parameters, atomic den-sity and interaction strength. We have verified this be-havior for a wide range of parameters, some of whichare exemplified in Fig.4 for γ = γ = 0 (Fig.4a). In- deed, all data points collapse on a single curve (Fig.4b)described by ˜ χ = f bl f bl . (11)This simple formula nicely illustrates the effects of excita-tion blocking on the optical susceptibility of the Rydberg-EIT medium: For f bl < χ ( ∞ )12 for f bl & f bl ∼ Ω for weak Rydberg excitation [22] this implies afinite cubic nonlinearity, as observed in [9].In conclusion, we have presented a numerical methodthat permits to explore the optical response of an EITmedium in the presence of strong atomic interactions. Inparticular, we considered, Rydberg-Rydberg atom inter-actions, which yield strong photon-photon interactions,manifested in a nonlinear Ω -dependence of the opticalsusceptibility. At small atomic densities our Monte Carloresults are in excellent agreement with reduced-densitymatrix calculations, but in addition are shown to be ap-plicable over a much wider range of densities and inter-action strengths. Beyond the present calculations, thedescribed approach permits efficient simulations of verylarge atom numbers, which in future studies should en-able detailed comparisons with experiments. Besides illu-minating the effect of Rydberg interactions, the revealeduniversal behavior of the probe absorption may be of usein experiments, by permitting quick estimates of opticalnonlinearities and allowing to determine the medium ab-sorption from pure population measurements. While wehave primarily focussed on nonlinear absorption phenom-ena the developed approach can also be used to study theoptical response of far-detuned EIT media and, thus, toexplore prospects for realizing strong coherent photon-photon interactions. [1] M. Fleischhauer, A. Imamoglu, and J.P. Marangos, Rev.Mod. Phys. , 633 (2005).[2] S. E. Harris, J. E. Field, and A. Imamoglu, Phys Rev.Lett. , 1107 (1990).[3] M. D. Lukin and A. Imamoglu, Phys. Rev. Lett. , 1419(2000).[4] M. Bajcsy et al., Phys. Rev. Lett. , 203902 (2009).[5] A.K. Mohapatra, T.R. Jackson, and C.S. Adams, Phys.Rev. Lett. , 113003 (2007).[6] A.K. Mohapatra et al., Nature Phys. , 890 (2008).[7] K.J. Weatherill et al., J. Phys. B , 201002 (2008).[8] U. Raitzsch et al., New J. Phys. , 055014 (2009).[9] J.D.Pritchard et al., Phys. Rev. Lett. 105, 193603 (2010)[10] R. Fleischhaker, T.N. Dey, and J. Evers, Phys. Rev. A , 013815 (2010).[11] H. Schempp et al., Phys. Rev. Lett. , 173602 (2010).[12] H. Weimer, et al., Phys. Rev. Lett. , 250601 (2008).[13] T. Pohl, E. Demler, and M.D. Lukin, Phys. Rev. Lett. , 043002 (2010). [14] J. Schachenmayer, I. Lesanovsky, A. Micheli and A.J.Daley, New J. Phys. , 063001 (2004); K.Singer et al., Phys. Rev. Lett. , 163001 (2004); T. Vogtet al., Phys. Rev. Lett. , 083003 (2006); R. Heidemannet al., Phys. Rev. Lett. 99, 163601 (2007).[16] M.D. Lukin et al., Phys. Rev. Lett. , 037901 (2001).[17] D. Moller, L.B. Madsen, and K. Molmer, Phys. Rev. Lett. , 170504 (2008).[18] M. M¨uller et al., Phys. Rev. Lett. , 170502 (2009).[19] L. Allen, and J.H. Eberly, Optical Resonance and Two-level Atoms (Wiley, New York, 1975).[20] L.R. Wilcox and W.E. Lamb, Phys. Rev. , 1915(1960).[21] C. Ates et al., Phys. Rev. A , 013413 (2007).[22] N. Henkel, R. Nath, and T. Pohl Phys. Rev. Lett.104