Nonlocal Nonlinear Optics in cold Rydberg Gases
NNonlocal Nonlinear Optics in cold Rydberg Gases
S. Sevin¸cli, N. Henkel, C. Ates, and T. Pohl Max Planck Institute for the Physics of Complex Systems, 01187 Dresden,Germany School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
We present an analytical theory for the nonlinear optical response of a strongly interacting Ry-dberg gas under conditions of electromagnetically induced transparency. Simple formulae for thethird order optical susceptibility are derived and shown to be in excellent agreement with recentexperiments. The obtained expressions reveal strong nonlinearities, which in addition are of highlynonlocal character. This property together with enormous strength of the Rydberg-induced non-linearities is shown to yield a unique laboratory platform for nonlinear wave phenomena, such ascollapse-arrested modulational instabilities in a self-defocussing medium.
PACS numbers: 32.80.Ee, 42.50.Gy,42.65.-k
Advances in designing materials with highly intensity-dependent refraction [1–3] have ushered in numerousstudies of nonlocal nonlinear wave phenomena [4–8].Many of these settings, such as, nematic liquid crystals[1, 2] or thermal media [3], require high power laser light.On the other hand, electromagnetically induced trans-parency (EIT) in ultracold multi-level atoms [9, 10] pro-vides an elegant mechanism to suppress photon loss andsimultaneously increase light-matter interaction times toenhance nonlinear effects. Combined with sufficientlylarge nonlinearities, this holds great potential for few-photon nonlinear optics [11, 12] and may enable appli-cations in communication and quantum information sci-ence.Recently, it was recognized that EIT-schemes involvinghighly excited atomic Rydberg levels provide promisingperspectives for such applications [13–22]. In particular,the huge polarizability of Rydberg states gives rise togiant Kerr coefficients [15], but also entails strong long-range interactions, which render Rydberg-EIT media in-trinsically nonlinear. Indeed, a recent theory for two-photon pulses revealed the emergence of strong effectivephoton-photon interactions [23], while experiments [21]and numerical calculations [24] demonstrated greatly en-hanced nonlinear absorption coefficients in the oppositelimit of large photon numbers.In this letter, we develop an analytical theory forthe nonlinear optical response of a strongly interactingRydberg-EIT medium to monochromatic multi-photonlight sources. Based on the approach, we give a simpleformula for the nonlinear absorption coefficient that pro-vides an excellent description of recent measurements oncold Rubidium gases [21]. For large single-photon detun-ings, absorption is shown to be greatly suppressed – yetmaintaining huge refractive nonlinearities, that exceedprevious records in ultracold Kerr media [10] by severalorders of magnitude. Combined with their long rangethis makes for an ideal nonlinear medium to study non-local wave phenomena, in which the strength , the range and even the sign of the nonlocal interaction kernel canbe widely tuned with high accuracy. To demonstrate this ¯¯¯¯ y [ µ m ] x [ µ m ] z [ µ m ] γ | !| !| ! (a) (b)(c) | !| ! Ω c Ω p Ω p Ω c ∆ / FIG. 1: a) Three-level scheme for isolated atoms, where theatomic ground state | (cid:105) , an intermediate state | (cid:105) and a highlyexcited Rydberg state | (cid:105) are mutually driven by a strong con-trol and a weak probe field with Rabi frequencies Ω c and Ω p ,respectively. On two-photon resonance, EIT ensures losslesspropagation of the optical fields, unaffected by spontaneousdecay ( γ ) and the single-photon detuning ∆ /
2. (b) In a gasof atoms, the strong van der Waals interaction between atomsin Rydberg states ( | (cid:105) ) inhibit multiple Rydberg excitationswithin a blockade radius R c , giving rise to a strongly non-linear optical response of the medium. The resulting non-linear beam propagation, for example, leads to modulationinstabilities, as shown in (c) for a Rubidium 70 S / Rydberggas with a density of 8 × cm − and Ω p / π = 0 . c / π = 80MHz, ∆ / π = 1 . potential, we present numerical results for the propaga-tion of cw laser light and show that paradigm phenom-ena, such as optical solitons and modulational instabili-ties (see Fig.1c) are observable with current experimentalcapabilities.Consider first the propagation of a beam with ampli-tude Ω p (see Fig.1b) and wavenumber k as described bythe paraxial wave equation (cid:18) − i k ∇ ⊥ + ∂∂z (cid:19) Ω p ( r ) = ik χ ( r )Ω p ( r ) , (1)where ∇ ⊥ accounts for the transverse dynamics with re-spect to the axial coordinate r ⊥ = ( x, y ) perpendicular a r X i v : . [ phy s i c s . a t o m - ph ] J un to the propagation direction z . The relevant mediumproperties are contained in the complex susceptibility χ = χ R + iχ I = 2 ℘ (cid:126) (cid:15) Ω p ρ , (2)which is determined by the dipole matrix element ℘ ofthe probe transition and the corresponding atomic coher-ence density ρ . The probe field, Ω p , drives the lowertransition between the ground state, | (cid:105) , and a low-lyingexcited state, | (cid:105) , of ladder-type three-level atoms (seeFig.1a), whose optical response is controlled by a strongcontrol field, driving the upper transition between | (cid:105) anda Rydberg state, | (cid:105) , with a Rabi frequency Ω c > Ω p .Without interactions, this yields a perfect EIT medium,in which each of the N atoms in the gas settles into adark state | d i (cid:105) ∝ Ω c | i (cid:105) − Ω p ( r i ) | i (cid:105) ( i = 1 , ..., N ) suchthat ρ = χ = 0 and the probe beam is unaffected by theatomic medium [9]. In the presence of strong Rydberg-Rydberg atom interactions the gas dynamics becomeshighly correlated due to the resulting level shifts of mul-tiply excited Rydberg states. Within a critical blockaderadius R c all but a single Rydberg excitation are inhib-ited [25] (see Fig.1b) and removed from two-photon reso-nance, thereby diminishing EIT, and, thus giving rise tononlocal absorption and refraction within a range ∼ R c .Since the Rydberg state population in the unperturbeddark states | d i (cid:105) is proportional to Ω p ( r i ) one, hence,expects an intensity-dependent, i.e. nonlinear, opticalresponse.Having established a simple picture of the basic mech-anisms we now derive the resulting optical susceptibilityfrom the underlying Heisenberg equations for the atomictransition operators ˆ σ ( i ) αβ = | α i (cid:105)(cid:104) β i | ( α, β = 1 , , p ( r i ) (cid:28) Ω c ) thesecan be expanded in Ω p / Ω c [26]. Upon adiabatic elimina-tion of ˆ σ ( i )12 one obtains a single dynamical equation forthe two-photon transition operator of the i th atom ddt ˆ σ ( i )13 = − Ω c Ω p ( r i ) + Ω c ˆ σ ( i )13 − γ σ ( i )13 − i (cid:88) j (cid:54) = i V ij ˆ σ ( j )33 ˆ σ ( i )13 , (3)where ˆ σ ( i )33 = ˆ σ ( i )31 ˆ σ ( i )13 , Γ = γ + γ − i ∆, ∆ / γ , γ and γ accountfor the spontaneous decay of the intermediate state aswell as the linewidth of the probe and two-photon tran-sition, respectively. The last term in eq.(3) describes theinteractions between atoms in the Rydberg state | (cid:105) and V ij = C / | r i − r j | denotes the corresponding van derWaals potential for atoms at positions r i and r j . Sincethe van der Waals coefficient C ∝ n drastically in-creases with the atom’s principal quantum number n ,the interaction between highly excited Rydberg atomsexceeds that of the two low-lying states by many ordersof magnitude. Proper inclusion of the resulting strongatomic correlations requires knowledge of the two-body Ω p0 / Ω c − l n ( T ) Ω p0 / Ω c ρ = 3 . × cm − c π = 3 . ρ = 1 . × cm − c π = 4 . FIG. 2: Nonlinear transmission of a cold Rubidium Rydberg-EIT medium with | (cid:105) = | S / (cid:105) at two different densitiesand control Rabi frequencies, and for γ / π = 110kHz and γ / π = 220kHz [21]. Up to Ω p0 ≈ . c there is goodagreement between our low-Ω p prediction eqs.(5) and (6)(solid line) and the experimental data [21, 27] (symbols). Thedashed lines neglect the drop in absorption due to attenuationand averaging over the initial transverse beam profile. correlators ˆ σ ( j ) αβ ˆ σ ( i ) α (cid:48) β (cid:48) whose dynamics follows from eq.(3)by applying the chain rule. Being primarily interestedin the leading order nonlinear contribution to χ , we canonce more expand the resulting two-body equations toleading order in Ω p . This amounts to dropping directthree-body correlators and, thus, yields a closed set ofevolution equations for the one- and two-body opera-tors. Setting ddt (cid:104) ˆ σ ( i ) αβ (cid:105) = ddt (cid:104) ˆ σ ( j ) αβ ˆ σ ( i ) α (cid:48) β (cid:48) (cid:105) = 0, the steadystate expectation values are then readily obtained fromthe resulting set of algebraic equations. Finally, wetake the continuum limit by defining continuous densi-ties ρ αβ ( r ) = (cid:80) i (cid:104) ˆ σ ( i ) αβ (cid:105) δ ( r − r i ), and obtain ρ ( r ) = iγ Ω p ( r )Ω + γ Γ ρ − Ω p ( r )Ω (Ω − γ Γ) | Ω + γ Γ | ρ × (cid:90) d r (cid:48) | Ω p ( r (cid:48) ) | V ( r − r (cid:48) )Ω + γ Γ + i Γ V ( r − r (cid:48) ) , (4)where ρ = (cid:80) i δ ( r − r i ) is the total atomic density. To-gether with eq.(2) this yields the leading-order nonlinearsusceptibility and permits to propagate the probe beamaccording to eq.(1).If the atoms are driven on single-photon resonance,∆ = 0, the main interaction effect will be nonlinear ab-sorption. Hence, one can neglect the transverse beamdynamics ( ∇ ⊥ ) as well as the nonlocality in eq.(4), bysetting Ω ( r (cid:48) ) ≈ Ω ( r ). With this simplification one ob-tains local first and third order susceptibilities, defined by χ ( r ) = χ (1) + χ (3) Ω ( r ). The remaining spatial integralin eq.(4) can be carried out analytically to give χ (1)R = 0 , χ (1)I = 6 πγγ k ( γ Γ + Ω ) ρ , (5) χ (3)R = − √ π γ Ω C | C | − / k √ Γ[ γ Γ + Ω ] / ρ , χ (3)I = | χ (3)R | . U U MI M I g r o w t h r a t ee ff ec t i v e p o t e n t i a l r ⊥ /R c k · R c (a) (b) ¯Ω α =50 ¯Ω α =80¯Ω α =20¯Ω α =1 FIG. 3: (a) Effective photon-photon interaction potentials,introduced in eq.(8). (b) Growth rate Γ MI of intensity mod-ulations with wavenumber k for defocussing nonlinearities ofdifferent strengths Ω α . The dashed lines show the corre-sponding imaginary part while the solid lines correspond tothe real part of Γ MI . The critical value of Ω α MI = 50 . α = 80. This expression permits a simple interpretation, by in-troducing the resonant blockade radius ˜ R c , defined bythe distance at which the interaction | C | / ˜ R exceedsthe width ˜ δ EIT = Ω / Γ of the EIT window [23]. Sub-stitution of C by ˜ R c = ( | C | / ˜ δ EIT ) / shows that χ (3) is proportional to the corresponding two-level responsetimes the number R ρ of blockaded atoms, which is con-sistent with the simple picture outlined above and thenumerical findings of [24].Experimentally, nonlinear absorption has been re-cently studied in a cold Rubidium gas involving | (cid:105) = | S / (cid:105) Rydberg states [21]. In the experiments thetransmission, T , of a Gaussian probe beam (Ω p =Ω p0 e − r ⊥ /w ) through the gas of length l was measuredfor different intensities and atomic densities. Within thelocal approximation this configuration permits a simplesolution of eq.(1) for the integrated beam transmission T = T ln(1 + p ) p , (6)where p = Ω χ (3)I (1 − T ) /χ (1)I and T = e − kχ (1) l isthe first order transmission. Fig.2 shows a comparisonto the measured transmission for two different densitiesand demonstrates good agreement, even for rather largeprobe Rabi frequencies of up to Ω p0 ≈ . c . Note thatthe backaction of the nonlinear beam attenuation ontosusceptibility and, equally important, the averaging overthe transverse beam profile are both essential for a properdescription of the experiment. Neglecting these effectsyields the dashed lines in Fig.2, which significantly over-estimates the nonlinear absorption.Since on resonance χ (3)I = | χ (3)R | [cf. eq.(5)], large non-linear refraction is inevitably accompanied by high pho-ton loss. However, for large single-photon detunings ∆ (cid:29) γ eq.(4) yields χ I ∼ ( γ/ ∆) χ (3)R , such that dissipative loss can be greatly suppressed. For instance, for a RubidiumRydberg gas with Ω c / π = 5MHz, Ω p / π = 0 . ρ = 8 × cm − one obtains a largeabsorption length of l abs ≈ n ≈ × cm /W which is 5 ordersof magnitude greater than previously obtained with ul-tracold Rb groundstate atoms at the same density [10].As refraction starts to dominate absorption, the nonlo-cality of χ (3) [cf. eq.(4)] becomes significant. To accountfor its effects on the transverse beam propagation we re-cast eqs.(2,4) into χ ( r ) = − πγρ k ∆Ω (cid:90) d r (cid:48) | Ω p ( r (cid:48)⊥ , z ) | | r (cid:48) − r | R − i γ ∆ | Ω p ( r (cid:48)⊥ , z ) | (cid:104) | r (cid:48) − r | R (cid:105) , (7)where we assumed γ (cid:28) γ , γ (cid:28) δ EIT = Ω / ∆and introduced the off-resonant blockade radius R c =( C /δ EIT ) / ( C ∆ >
0) [23] set by the off-resonantEIT width δ EIT . To simplify matters, we proceed bydefining scaled coordinates τ = z/ ( kR ), ξ = r ⊥ /R c and the dimensionless probe amplitude Ω, normalized to (cid:82) Ω ( ξ , τ ) d ξ = 1. Retaining the local approximationalong the propagation direction [31] this yields a two-dimensional nonlinear Schr¨odinger equation i∂ τ Ω( ξ , τ ) = (cid:34) − ∇ ξ α (cid:90) d ξ (cid:48) | Ω( ξ (cid:48) , τ ) | U ( ξ − ξ (cid:48) ) (8) − i γ ∆ α (cid:90) d ξ (cid:48) | Ω( ξ (cid:48) , τ ) | U ( ξ − ξ (cid:48) ) (cid:21) Ω( ξ , τ ) , where α = π ρ γ P p (cid:126) k cR Ω C parametrizes the strength ofthe nonlinearity, P p denotes the probe beam powerand the effective interaction potentials U m ( ξ ) = (cid:82) ∞−∞ d z (cid:2) ξ + z ) (cid:3) − m are shown in Fig.3a. As α ∝ C , repulsive atomic interactions lead to self-defocussingnonlinearities, while attractive atomic interactions maponto self-focussing nonlinearities.The former case, can, e.g., be realized with coldRb( nS / ) Rydberg states as in the experiments [21] -9 -6 -3 0 3 6 9-9 -6 -3 0 3 6 9-9 -6 -3 0 3 6 9-9-6-30369 x/R c y / R c x/R c x/R c (a) (b) (c) | Ω | | Ω max | FIG. 4: Output beam profile | Ω | ( r ⊥ ) for Ω p0 / π = 0 . c / π = 80MHz and ∆ / π = 1 . l = 210 µ m through a Rb(70 S / ) EIT medium at three dif-ferent densities (a) 4 × cm − , (b) 5 . × cm − and (c)8 × cm − . For the color coding each distribution has beennormalized by the actual maximum intensity | Ω max | . -3-2-10123 x/R c y / R c x/R c (c) (e) -3 -2 -1 0 1 2 3-3-2-1012 -2 -1 0 1 2 3 y / R c (d) (f)(a)(b) x / R c x / R c y / R c y / R c | Ω | / | Ω m a x | | Ω | / | Ω m a x | FIG. 5: Intensity profiles | Ω | ( r ⊥ ) for a Sr(50 S ) EITmedium with Ω / π = 0 . c / π = 15MHz, ∆ / π =3 . × cm − and(b,d,f) 1 . × cm − . Panels (a,b) show the stable solitonsolutions, while panels (c-f) show the compressed output in-tensity profile after a propagation length l = 240 µ m for aninput beam with (c,d) ν = 2 and (e,f) ν = 6. The color codingis identical to Fig.4. discussed above. While the resulting photon-photon in-teractions are isotropically repulsive, the correspondingmomentum space interaction ˜ U ( k ) is not sign-definite.Despite being defocussing , the present nonlocal inter-action can, consequently, promote a modulational in-stability. In Fig.3b we show the corresponding rateΓ MI ( k ) = − k (cid:113) k − α Ω ˜ U ( k ) [4] for a given mode withwave number k to grow out of a homogenous amplitudeΩ( ξ ) = const . . This growth rate Γ MI assumes real valueswithin a narrow range around k ≈ π at a critical interac-tion strength α Ω ≈
50 [28], resulting in stable transverseintensity modulations on a length scale ∼ R c as the beampropagates through the medium (cf. Fig.1c). To examinetheir observability, we performed numerical simulationsof eq.(8) for a Rb(70 S / ) Rydberg gas traversed by asuper-Gaussian beam Ω = Ω e − ( ξ/w ) ν + iφ with ν = 6 andsmall spatial phase noise φ . Fig.4 shows calculated out-put intensity profiles for different atomic densities anddemonstrates that highly localized, rather regular inten-sity patterns can be realized in high density gases withfeasible laser parameters.Self-focussing nonlinearities arise from attractive Ry-dberg interactions, as occurring between n S states ofStrontium atoms [29], for which EIT has been recentlyobserved [14]. In this case, modulational instabilities can,in principle, occur for any α < stable bright solitons, leading to tight beam focussing.A simple variational analysis of eq.(8) (see, e.g., [30])yields a critical interaction strength of α so ≈ .
71, abovewhich stable bright solitons exist. Figs.5a and 5b showtwo examples for a Sr(50 S ) gas and reveal a character- istic soliton size (cid:46) R c . In addition we show final inten-sity profiles for a Gaussian ( ν = 2) and super-Gaussian( ν = 6) input beam. All cases are for an input width of w = 3 R c and demonstrate significant focussing after theconsidered propagation length l = 240 µ m. In the lattercase, the beam compression is superimposed by a radialMI leading to a ring-shaped hollow output beam. Witha typical size of several µ m these structures are readilyobservable experimentally.In conclusion, we have presented a theory for the non-linear response of a strongly interacting Rydberg-EITgas, giving good agreement with recent measurements.The derived expressions for the third-order susceptibil-ity suggest that huge nonlinearities of highly nonlocalcharacter can be experimentally realized, which providesan ideal setting to study complex nonlinear wave phe-nomena. To demonstrate these prospects we have shownthat the observation of basic effects such as the forma-tion of bright solitons and collapse-arrested modulationalinstabilities are within experimental reach. The latteris particularly interesting in the uncommon case of de-focussing, nonlocal nonlinearities. Here, the repulsionbetween emerging intensity peaks combined with trans-verse beam confinement may promote the formation oftransverse supersolid or crystalline states of photons.This question may be addressed within the present ap-proach, extended to quantum light in order to accountfor atom-photon and photon-photon correlations, whichwould open up a general framework for studying many-body physics with strongly interacting photons. Froma different perspective, we expect the discussed nonlin-ear light propagation to be relevant for interpreting coldRydberg gas experiments at high densities.We are grateful to A. Gorshkov, M.D. Lukin,J.D. Pritchard, C.S. Adams, S. Skupin, J. Otterbachand M. Fleischhauer for valuable discussions, and thankC.S. Adams for providing unpublished experimentaldata. [1] M. Peccianti et al. , Optics Letters, et al. , Phys. Rev. Lett. , 113902 (2004).[3] C. Rotschild et al. , Nature Phys.
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