Blockade-induced resonant enhancement of the optical nonlinearity in a Rydberg medium
Annika Tebben, Clément Hainaut, Valentin Walther, Yong-Chang Zhang, Gerhard Zürn, Thomas Pohl, Matthias Weidemüller
BBlockade-induced resonant enhancement of the optical nonlinearity in a Rydbergmedium
Annika Tebben, Clément Hainaut, Valentin Walther, Yong-ChangZhang, Gerhard Zürn, Thomas Pohl, and Matthias Weidemüller
1, 3 Physikalisches Institut, Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany Department of Physics and Astronomy, Aarhus University,Ny Munkegade 120, DK 8000 Aarhus C, Denmark National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,and CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics,Shanghai Branch, University of Science and Technology of China, Shanghai 201315, China (Dated: September 17, 2019)We predict a resonant enhancement of the nonlinear optical response of an interacting Rydberggas under conditions of electromagnetically induced transparency. The enhancement originatesfrom a two-photon process which resonantly couples electronic states of a pair of atoms dressedby a strong control field. We calculate the optical response for the three-level system by explicitlyincluding the dynamics of the intermediate state. We find an analytical expression for the thirdorder susceptibility for a weak classical probe field. The nonlinear absorption displays the strongestresonant behavior on two-photon resonance where the detuning of the probe field equals the Rabifrequency of the control field. The nonlinear dispersion of the medium exhibits various spatialshapes depending on the interaction strength. Based on the developed model, we propose a realisticexperimental scenario to observe the resonance by performing transmission measurements.
I. INTRODUCTION
A Rydberg gas under conditions of electromagneticallyinduced transparency (EIT) exhibits a nonlinear opticalresponse, which exceeds that of conventional media byorders of magnitude [1, 2] . Aiming at the full controlof effective photon interactions, numerous experimentalachievements, such as the realization of single-photontransistors [3, 4] and the creation of bound states of pho-tons [5, 6], as well as advanced theoretical investigationsboth in the quantum [7–11] and semi-classical regimes[12–15] have been reported.In the quantum regime, the notion of dark-state po-laritons has proven to be successful for the theoreticaldescription of photon propagation through an interact-ing Rydberg medium [7, 8]. In the case of two inter-acting photons a wavefunction approach was developed[9] and allowed to accurately describe the experimentalfindings of dissipative [16], spin-exchange like [17] as wellas attractive photonic interactions [5, 6]. More complexmodels to describe the photon propagation investigatedthe scattering properties of two polaritons [10] and madethe transition to the few- and many-body regime in onedimension utilizing an effective field theory [11].In the semi-classical regime, a Monte Carlo rate equa-tion model was used to obtain an expression for the non-linear response of the atomic gas by including Rydberginteractions as level shifts [12] or by using a superatomapproach [13]. This picture was condensed to a univer-sal scaling of the nonlinear absorption with the fractionof Rydberg blockaded atoms. This scaling proved tobe consistent with calculations in the quantum regime,that are typically much more complicated. Moreover,it showed excellent agreement with experimental results [14], underlining the strength of this basic model. Dueto the long-range interactions between Rydberg atoms,the nonlinearity in Rydberg-EIT systems is intrinsicallynonlocal. Based on a cluster expansion, an analytic ex-pression for this nonlocal optical response of a Rydberggas has been derived [15]. These results proved the ex-istence of modulational instabilities, which are a precur-sor of photon crystallization. All these semi-classical ap-proaches neglect the dynamics of the intermediate state.However, including these dynamics revealed interestingcharacteristics of the photonic and atomic pair potentials[10, 18, 19].Here, we develop a semi-classical model for the nonlo-cal, nonlinear response of an interacting Rydberg gas, ex-plicitly including the dynamics of the intermediate state.We reveal the existence of a two-body, two-photon reso-nance in the optical response when the control field Rabifrequency is tuned to the probe field detuning.In order to provide a simple picture, we start by de-scribing the system based on a pair-state model and ex-plain how atomic interactions lead to a two-photon res-onance. We then derive an analytical expression for thenonlinear response of the interacting Rydberg gas for ar-bitrary interaction strengths starting from the Maxwell-Bloch equations. We show that in the presence of theresonance the nonlinear response can be significantly en-hanced. We discuss the spatially dependent absorptionfeatures of the nonlinear response and present the scalingof the enhancement with relevant field and atom param-eters. Finally, we propose a feasible transmission mea-surement revealing the resonance. a r X i v : . [ phy s i c s . a t o m - ph ] S e p Figure 1. (a) Atomic level structure of Rydberg atoms, whichinteract via an interaction V ( R ) , that depends on the inter-atomic distance R . (b) Relevant level scheme in the dressedpair-state basis, as explained in the main text. For ∆ = − Ω c the eigenstate | β − (cid:105) moves into two-photon resonance with theground state | gg (cid:105) . II. LASER-DRESSED INTERACTINGPAIR-STATES
Consider a ladder-type realization of the EIT scheme,where a gas of Rydberg atoms with density ρ is exposedto counter-propagating probe and control fields as shownin Fig. 1(a). The coherent probe field E ( r , t ) with fre-quency ω p and Rabi frequency Ω p couples the atomicground state | g (cid:105) to a short lived intermediate state | e (cid:105) with decay rate γ e , while a control field with Rabi fre-quency Ω c drives the transition to a metastable Rydbergstate | r (cid:105) with a small decay rate γ r . The two-photondetuning δ for the ground to Rydberg state transition iskept at zero, but the fields are detuned from the inter-mediate state by the single-photon detuning ∆ , as shownin Fig. 1(a).The system is governed by pairwise van der Waalsinteractions V ( R ) , giving rise to the so-called Rydbergblockade effect. Here, two atoms at a distance smallerthan the blockade radius R b cannot simultaneously beexcited to the Rydberg states [20]. In the case of Ry-dberg EIT, R b is defined as the distance where the vander Waals potential exceeds the EIT linewidth δ EIT =Ω c / | γ e − i ∆ | [21]. Thus, R b = ( c /δ EIT ) / is the char-acteristic length scale of the system.Considering pair-wise interactions, it is natural to ex-amine the coupled atom-light system in the pair-state basis. The corresponding Hamiltonian [18] ˆ H = √ p √ p − ∆ Ω c √ p c p √ p − √ c
00 0 Ω p √ c − ∆ √ c √ c V ( R ) (1)describes the coupling between the ground state | gg (cid:105) andthe states {| ge (cid:105) + , | gr (cid:105) + } and {| ee (cid:105) , | er (cid:105) + , | rr (cid:105)} in thesingly- and doubly excited subspaces, respectively. Here,we make use of the symmetric pair-state basis, where | ij (cid:105) + = ( | ij (cid:105) + | ji (cid:105) ) / √ with i, j ∈ { g, e, r } .In the limit of vanishing interactions ( V ( R ) → ) atlarge interatomic distances, the system reduces to a gasof individual atoms under EIT conditions, featuring alinear response to the applied fields [21].In the following, we discuss how the presence of inter-actions changes the energy spectrum of the eigenstates of ˆ H . For Ω c (cid:29) Ω p the singly- and doubly excited subspacescan be dressed by the control field individually [18], lead-ing to eigenstates {| α (cid:105) + , | α (cid:105) − } and {| β (cid:105) − , | β (cid:105) + , | β (cid:105) } ,respectively, as shown in Fig. 1(b).In the limit of strong interactions ( V ( R ) (cid:29) Ω c ), theeigenstate | β (cid:105) mainly contains the doubly excited Ry-dberg state and is decoupled from the remaining levelsystem, as schematically shown in Fig. 1(b). Here, theground state | gg (cid:105) is coupled by two probe photons tothe dressed states of the doubly-excited subspace. Thiscoupling becomes maximal for Ω c = ± ∆ , as shown inFig. 2(a), and establishes a two-body, two-photon reso-nance, that has already inspired the method of resonantRydberg dressing [18, 19].In the case of finite interactions, the influence of thedoubly excited Rydberg state | rr (cid:105) on the energy spec-trum has to be considered explicitly. Here, the dressedstate | β (cid:105) alters the energy spectrum and shifts thestates | β ± (cid:105) to lower energies as shown exemplarily for V ( R = 2 . µ m) and ∆ < in the right graph of Fig.2(a). For ∆ > this happens in a similar manner, suchthat we only display one case here for clarity. As a resultof these energy shifts, the ratio where the two-photon res-onance condition is met shifts to smaller values, in thisexample to | Ω c / ∆ | = 0 . .Fig. 2(b) highlights this effect and shows the reso-nance position | Ω c / ∆ | res against the inter-atomic sepa-ration R , meaning different interaction strengths. Forevery R there exists exactly one ratio | Ω c / ∆ | for nega-tive (black) and positive (green) single-photon detunings,where the resonance condition is met.Considering the propagation of the probe field, thisresonance changes the nonlinear optical response of theRydberg gas, for which we will derive an analytical ex-pression in the following. Figure 2. (a) Energy ∆ E of the dressed levels | β − (cid:105) (solidline), | β + (cid:105) (dashed line) and | β (cid:105) (dashed-dotted) against theratio | Ω c / ∆ | for positive (green) and negative values (black)of ∆ , respectively. ∆ E = 0 corresponds to the two-photonresonance with the ground state, which is met for infinite( R → , left) and finite (right) interactions for different | Ω c / ∆ | . (b) Resonance position | Ω c / ∆ | res against the inter-atomic distance R for positive (green) and negative (black)single-photon detunings. III. NONLINEAR OPTICAL RESPONSE
In this section, we derive a spatially dependent analyt-ical expression for the nonlinear, nonlocal susceptibilityof the Rydberg EIT gas, that allows to study the op-tical response for various interaction strengths, non-flatprobe fields and non-constant atomic density distribu-tions. For this purpose, we first introduce a set of bosonicMaxwell-Bloch equations that accurately describe the in-teracting many-body system under weak-driving condi-tions. Next, we proceed by solving these equations fora classical probe field exactly up to the third order ina cluster expansion. Finally, we discuss the spatially-dependent refraction and absorption features of the non-linear, nonlocal susceptibility.
A. Maxwell-Bloch equations
The bosonic Maxwell-Bloch equations for the Rydberg-EIT system read [2] ∂ t E ( r ) = (cid:18) ic ∇ ⊥ k p − c∂ z (cid:19) E ( r ) − ig √ ρ ˆ P ( r ) , (2) ∂ t ˆ P ( r ) = − ig √ ρ E ( r ) − i Ω c ( r ) ˆ S ( r ) − Γ e ˆ P ( r ) , (3) ∂ t ˆ S ( r ) = − i Ω c ( r ) ˆ P ( r ) − Γ r ˆ S ( r ) − i (cid:90) d r (cid:48) V ( r − r (cid:48) ) ˆ S † ( r (cid:48) ) ˆ S ( r (cid:48) ) ˆ S ( r ) , (4)where we dropped the time-dependence of the fields andoperators for convenience. Eq. (2) describes, in paraxial approximation, the prop-agation of a classical probe field E ( r , t ) in z -directionthrough a medium with source term − ig √ ρ ˆ P ( r ) . Here, g √ ρ is the collectively enhanced single-atom couplingstrength g of the probe transition, k p = ω p /c thewavenumber of the probe field, c the speed of light, and ˆ P ( r ) a bosonic operator for the polarisation coherenceas motivated below. The assumption of a classical probefield is meaningful, if photon-photon and photon-atomcorrelations can be neglected, implying that the coher-ent nature of the field is preserved [2]. This is true, aslong as the atomic interactions and the coupling g tothe probe field are small. In the case of Rydberg-EIT,this is given for an optical depth per blockade radiusOD b ∝ gρc / (cid:28) [2].If the probe field is weak compared to the control field,the atomic part of the Maxwell-Bloch equations is reason-ably described in terms of continuous bosonic operators ˆ P ( r , t ) and ˆ S ( r , t ) for the polarisation and Rydberg spin-wave coherence, respectively [7, 8]. Moreover, withinthe weak-probe assumption, population decay can be ne-glected and only the coherence decay rates ¯ γ e,r = γ e,r / remain. We defined Γ e = ¯ γ e − i ∆ and Γ r = ¯ γ r − iδ .Eq. (2) to (4) have been solved in the semi-classicalregime for ∆ or γ e (cid:29) Ω c [2, 15], where the intermediatestate dynamics can be eliminated. In these works, ithas been shown, that the Rydberg EIT system exhibitsa strong nonlinear and nonlocal response to the drivingfield. Motivated by this, we recast, in steady-sate, Eq.(2) into i∂ z E ( r ) = − ∇ ⊥ k p E ( r ) + χ (1) ( r ) E ( r )+ (cid:90) d r (cid:48) χ (3) ( r − r (cid:48) ) |E ( r (cid:48) ) | E ( r ) , (5)where the linear χ (1) ( r ) and nonlinear susceptibility χ (3) ( r − r (cid:48) ) are directly related to the polarisation co-herence via (cid:104) ˆ P ( r ) (cid:105) = cg √ ρ (cid:104) χ (1) ( r ) E ( r )+ (cid:90) d r (cid:48) χ (3) ( r − r (cid:48) ) |E ( r (cid:48) ) | E ( r ) (cid:21) . (6)In Eq. (5), the two complex susceptibilities given in Eq.(10) and (17) act as an effective light potential responsi-ble for refraction and absorption on the linear and non-linear level, respectively. B. Perturbative solution
For Ω p (cid:28) Ω c , we proceed by solving the Maxwell-Bloch equations with a perturbative expansion in theprobe field. For this purpose we separate the probe fieldas E ( r ) = E f ( r ) , where the position dependence is ab-sorbed in f ( r ) and E is a small parameter. We expandthe expectation values of the polarisation coherence interms of E as (cid:104) ˆ P ( r ) (cid:105) = P (0) ( r ) + E P (1) ( r ) + E P (2) ( r )+ E P (3) ( r ) + O ( E ) (7)and similarly for the spin-wave coherence ˆ S ( r ) . Insertingthis into Eq. (3) and (4) allows to solve the problemorder by order.In zeroth-order the probe field vanishes, such that allatoms remain in the ground state. Therefore, P (0) ( r ) = S (0) ( r ) = 0 . Moreover, the second- and all higher evenorders vanish due to the centro-symmetry of the atomicgas.The first-order has the solution P (1) ( r ) = − ig √ ρ Γ r Ω ( r ) + Γ r Γ e f ( r ) , (8) S (1) ( r ) = − g √ ρ Ω( r )Ω ( r ) + Γ r Γ e f ( r ) . (9)Inserting the result for P (1) ( r ) into Eq. (6) leads to thelinear susceptibility χ (1) ( r ) = − ig Γ r c (Ω c + Γ r Γ e ) ρ ( r ) . (10)It recovers the well-known effect of EIT in the absenceof atomic interactions and leads, for γ r = 0 , to a fulltransmission of the probe field on two-photon resonance( δ = 0 ).Solving the third-order equations ∂ t P (3) ( r ) = − i Ω( r ) S (3) ( r ) − Γ e P (3) ( r ) , (11) ∂ t S (3) ( r ) = − i Ω( r ) P (3) ( r ) − Γ r S (3) ( r ) − i (cid:90) d r (cid:48) V ( r − r (cid:48) ) (cid:104) ˆ S † ( r (cid:48) ) ˆ S ( r (cid:48) ) ˆ S ( r ) (cid:105) (12)is more involved due to the appearance of correlations be-tween Rydberg spin-wave excitations (cid:104) ˆ S † ( r (cid:48) ) ˆ S ( r (cid:48) ) ˆ S ( r ) (cid:105) .In the following, we explain the main steps of calculatingthis correlator.The time dependence of the Rydberg spin wave corre-lator ∂ t (cid:104) ˆ S † ( r (cid:48) ) ˆ S ( r (cid:48) ) ˆ S ( r ) (cid:105) = − i Ω c ( r ) (cid:104) ˆ S † ( r (cid:48) ) ˆ S ( r (cid:48) ) P ( r ) (cid:105) + i Ω c ( r (cid:48) ) (cid:104) (cid:104) ˆ P † ( r (cid:48) ) ˆ S ( r (cid:48) ) ˆ S ( r ) (cid:105) − (cid:104) ˆ S † ( r (cid:48) ) ˆ P ( r (cid:48) ) ˆ S ( r ) (cid:105) (cid:105) − [3¯ γ r + iV ( r − r (cid:48) )] (cid:104) ˆ S † ( r (cid:48) ) ˆ S ( r (cid:48) ) ˆ S ( r ) (cid:105)− i (cid:90) d r (cid:48)(cid:48) V ( r (cid:48) − r (cid:48)(cid:48) ) (cid:104) ˆ S † ( r (cid:48) ) ˆ S † ( r (cid:48)(cid:48) ) ˆ S ( r (cid:48) ) ˆ S ( r (cid:48)(cid:48) ) ˆ S ( r ) (cid:105) (13)is given by Eq. (3) and (4). As a two-body correlatorit requires knowledge of other two-body correlators asfor instance (cid:104) ˆ P † ( r (cid:48) ) ˆ S ( r (cid:48) ) ˆ S ( r ) (cid:105) , as well as the three-body correlator in the last line of Eq. (13). Ultimately thisleads to an infinite hierarchy of equations for the many-body system, that needs to be truncated appropriately.Here, the weak-probe assumption in combination withthe blockade effect provides a natural way of truncatingthe hierarchy as it limits the density of Rydberg exci-tations in the system [2]. Therefore, the probability offinding two Rydberg excitations within a blockaded vol-ume is small, and becomes negligible for three or moreexcitations. In this case, we can discard three-body inter-actions and correlations of this and higher orders are fullysuppressed [2]. This does not only allow to truncate thehierarchy of equations, but also implies that two-bodyatomic correlations are taken into account exactly.Applying this approach, we neglect terms as for ex-ample the last line of Eq. (13) and in a similar mannerobtain the time derivatives of all involved one- and two-body correlators. This leads to 20 coupled, linear equa-tions. In order to proceed with the calculation, we makethe ansatz ˆ P ( r ) = − ig √ ρ E f ( r ) Γ r Ω ( r ) + Γ r Γ e ˆ P (cid:48) ( r ) , (14) ˆ S ( r ) = − g √ ρ E f ( r ) Ω( r )Ω ( r ) + Γ r Γ e ˆ S (cid:48) ( r ) . (15)motivated by the first-order solutions of the Maxwell-Bloch equations given in Eq. (8) and (9). Performingthe associated variable change and assuming a spatiallyconstant Rabi frequency of the control field, makes theequations position-independent and allows to rephrasethem as a × -matrix in the steady-state. Solvingthe system gives the exact solution for the Rydberg spin-wave correlator (cid:104) ˆ S † ( r (cid:48) ) ˆ S ( r (cid:48) ) ˆ S ( r ) (cid:105) = − Ω c | a | r + Γ e ) g ρ ( r (cid:48) ) (cid:112) ρ ( r )2 a (Γ r + Γ e ) + i ( a + Γ e ) V ( r − r (cid:48) ) |E ( r (cid:48) ) | E ( r ) (16)up to two-body interactions, where we introduced theabbreviation a = Ω c + Γ r Γ e . Inserting this expression inthe third-order equations of the expansion finally leadsto the third-order susceptibility χ (3) ( r − r (cid:48) ) = Ω c g ρ ( r (cid:48) ) ρ ( r ) c | a | a × r + Γ e ) V ( r − r (cid:48) )2 a (Γ r + Γ e ) + i ( a + Γ e ) V ( r − r (cid:48) ) . (17)Having obtained a result for the first- and third-ordersusceptibility we arrive at a closed Eq. (5) for the prop-agation of the probe field through the highly nonlinearand nonlocal Rydberg EIT medium. Figure 3. Real (purple) and imaginary part (blue) of thenonlinear susceptibility χ (3) ( R ) against the inter-particle dis-tance R for various ratios Ω c / ∆ (left column: ∆ > , rightcolumn: ∆ < ). The susceptibility is scaled with a factor of g ρ / ( c Ω c ¯ γ e ) . Plotted with Ω c = 2 . γ e and | S / (cid:105) as theRydberg state of Rb atoms for a constant atomic densitydistribution with ρ = 2 × cm − . The blockade radii R b are { . , . , . } µ m for | Ω c / ∆ | = { . , . , } , respectively. C. Spatial shape of the nonlinearity
After having derived an analytic expression for thenonlinear, nonlocal susceptibility in Eq. (17), we are in aposition to investigate its spatially dependent absorptionand refraction features, given by its imaginary and realpart, respectively.Fig. 3 displays typical shapes of the nonlinear suscep-tibility χ (3) ( R ) as a function of the inter-atomic distance R for a constant atomic density distribution ρ ( r ) = ρ .For large R , the real and imaginary part tend to zerofor all ratios Ω c / ∆ , reflecting the trivial non-interactingregime. In the case of R → the real and imaginarypart are constant for a large range of atomic distances R with plateau values χ (3)0 , Re and χ (3)0 , Im , respectively. For Ω c = | ∆ | the latter gets maximal, meaning that the sys-tem displays the strongest nonlinear absorption.For intermediate atomic distances R the shape of thereal and imaginary part strongly depends on the ratio Ω c / ∆ and can display additional features. First, examinethe case for a positive single-photon detuning ( ∆ > ,left column). For Ω c / ∆ < , both the imaginary andreal part of the nonlinear susceptibility feature a soft-coreshape. However, for Ω c = ∆ the real part shows a strongmaximum and for Ω c > ∆ it features a sign-change wherethe imaginary part gets minimal at a finite distance. For ∆ < (right column in Fig. 3) the situation is reversed,such that the minimum of the imaginary part at a finite distance appears for Ω c < | ∆ | . Moreover, it is morepronounced than for ∆ > .The observed position of the additional features is adirect consequence of the van der Waals interactions andcan be understood in terms of the energies of the dressedeigenstates. Examining Fig. 2(b) we see, that for ∆ > (green) the resonance condition is only met for absolutevalues of the ratio Ω c / ∆ being larger than 1, while for ∆ < (black) the opposite holds. This is exactly thereason, why we observe a minimum of the imaginarypart of the nonlinear susceptibility for ratios Ω c / ∆ larger(smaller) than 1 for positive (negative) single-photon de-tunings in Fig. 3.As a result, the ratio Ω c / ∆ allows for spatial shaping ofthe absorption and refraction properties of the nonlinearsusceptibility. D. Scaling of the resonance
We now discuss the scaling properties of the resonanceby looking at the susceptibility for R → . Assuming ¯ γ r = 0 for simplicity we obtain χ (3)0 , Re = 2 g ρ c Ω c ∆ (cid:0) ¯ γ e + ∆ − Ω c (cid:1) ¯ γ e + (∆ − Ω c ) + 2¯ γ e (∆ + Ω c ) ≈ g ρ c ∆ , for Ω c = | ∆ | , ¯ γ e (cid:28) | ∆ | (18)for the real part and χ (3)0 , Im = − g ρ c Ω c ¯ γ e (cid:0) ¯ γ e + ∆ + Ω c (cid:1) ¯ γ e + (∆ − Ω c ) + 2¯ γ e (∆ + Ω c ) ≈ − g ρ c ¯ γ e ∆ , for Ω c = | ∆ | , ¯ γ e (cid:28) | ∆ | (19)for the imaginary part. Here, the second line in Eq. (18)and (19) gives the value at the resonance condition Ω c = | ∆ | in the non-adiabatic limit.Fig. 4(a) displays the real and imaginary part of χ (3)0 as a function of the ratio Ω c / ∆ . Here, the imaginarypart is resonantly enhanced for Ω c = | ∆ | , in agreementwith the discussion in the pair-state basis in section II.The real part exhibits a sign change with a negative slopearound Ω c / ∆ = ± .At the resonance condition Ω c = | ∆ | , the imaginarypart interestingly depends on the intermediate state de-cay rate, while the real part does not. This allows toincrease the imaginary part independently by choosingan atomic species with a long-lived intermediate state. IV. SIGNATURE OF THE RESONANCE INTHE PROBE TRANSMISSION
In this section, we investigate whether the two-body,two-photon resonance is experimentally accessible. For
Figure 4. Enhancement of the nonlinear susceptibility. Real(purple) and imaginary part (blue) of the nonlinear suscep-tibility χ (3)0 for strong interactions ( V ( R ) (cid:29) Ω c ) against theratio Ω c / ∆ . At the resonance position ( Ω c = | ∆ | ) the realpart features a sign change, while the imaginary part is reso-nantly enhanced. Plotted using Rb atoms with | S / (cid:105) asthe Rydberg state and Ω c = 2 . γ e . this purpose, we solve the propagation Eq. (5). Numeri-cally, this can be done in a straightforward manner by ex-ploiting a split-step Fourier propagation scheme [22, 23].However for a better understanding, we derive an an-alytic solution of the propagation Eq. (5) under the as-sumption of a flat input field. Neglecting diffraction, thisresults in an effective one-dimensional equation ∂ z I ( z ) = a I ( z ) + a I ( z ) (20)for the probe field intensity I ( z ) = |E ( z ) | , with a = 2 Im (cid:110) χ (1) (cid:111) , (21) a = 2 (cid:90) d r (cid:48) Im (cid:110) χ (3) ( r − r (cid:48) ) (cid:111) . (22)Eq. (20) holds if the probe field intensity is approxi-mately constant over the range of the nonlinear suscep-tibility (Fig. 3). This so called local approximation al-lows us to reduce the convolution integral in Eq. (5) toan integration solely over the susceptibility in Eq. (22).In addition, we assume in the simplest case a constantatomic density distribution. In this case only V ( r − r (cid:48) ) is left to be position dependent.A solution of Eq. (20) can be obtained readily andreads I ( z ) = a I e a z a + a I − a I e a z ≈ I e a z + a a e a z ( e a z − I + O ( I ) , (23)where the second line is an expansion for a small initialprobe field intensity I = I (0) . The first order describesan exponential reduction of the intensity, while the sec-ond contains the nonlinear absorption. Eq. (23) providesa leading-order nonlinear description of the probe field’spropagation in the limit of a flat input field and a con-stant intensity distribution of the control field. Figure 5. Transmission T = I ( L ) / I of the probe field af-ter propagating a distance L as a function of the single-photon detuning ∆ for Ω p / Ω c = 0 . , . , . , . , . (black to blue), respectively. In the nonlinear regime withhigh Ω p , two absorption minima for ∆ ≈ ± Ω c appear as aconsequence of the two-body, two-photon resonance. Plottedfor { Ω c , δ, ρ, L } = { γ e , , × cm − , µ m } using Eq.(23). Fig. 5 shows a transmission spectrum of the probe fieldas a function of the single-photon detuning. In the non-interacting regime (small Ω p ), the transmission equals1 for all ∆ , due to the EIT effect on two-photon reso-nance, where δ = 0 . Increasing the Rabi frequency of theprobe field gradually, the interacting, nonlinear regimeis reached. Here, two transmission minima occur as aconsequence of the enhanced susceptibility at ∆ ≈ ± Ω c .The overall shift of the spectrum towards negative val-ues of ∆ is a result of an integration over the nonlinearsusceptibility in Eq. (22), as the shape of its imaginarypart exhibits a minimum at a finite distance for a posi-tive (negative) ratio Ω c / ∆ above (below) 1, as shown inFig. 3.The distinct absorption features in the transmissionspectrum allow to access the resonance effect experimen-tally. However, for a realistic experimental situation aGaussian atomic density distribution should be consid-ered. This is straightforward as explained in AppendixA and the resonance is still observable. The parametersin Fig. 5, indicate that the two-body, two-photon reso-nance is experimentally accessible. V. CONCLUSION AND OUTLOOK
In conclusion, we predicted an enhancement of thenonlinear optical response of an interacting Rydberg gasunder EIT conditions. This enhancement is a conse-quence of an interaction-induced two-body, two-photonresonance. We developed a semi-classical theory in thenon-adiabatic, many-body regime in order to derive ananalytic expression for the nonlinear optical response forarbitrary interaction strengths, non-flat probe fields andnon-constant atomic density distributions. We showedthe enhancement as well as its scaling properties with rel-evant field and atom parameters. We demonstrated thatthe ratio of Ω c / ∆ can be used to tune the spatial depen-dence of the optical response pointing towards prospectsof shaping the effective light potential.In the quantum regime, a sign change of the effectivephotonic potential has been predicted [10] and indica-tions for an asymmetric behavior of the optical responsedepending on the sign of the detuning have been reported[5, 24]. Our work adds a semi-classical perspective toboth. Moreover, the derived scaling of the enhancementwith /γ e indicates that a highly nonlinear regime couldbe reached by using atoms with long lived intermediatestates as for example Strontium atoms. Our findings en-courage to investigate the yet unexplored non-adiabaticregime of Rydberg-EIT physics for low optical depth perblockade radius. ACKNOWLEDGMENTS
This work is part of and supported by the DFG Pri-ority Program "GiRyd 1929" (DFG WE2661/12-1), theDNRF through a Niels Bohr Professorship to T.P., theHeidelberg Center for Quantum Dynamics, the DFG Col-laborative Research Center "SFB 1225 (ISOQUANT)",and the European Union H2020 FET flagship projectPASQuanS (Grant No. 817482). A.T acknowledges sup-port from the Heidelberg Graduate School for Fundamen-tal Physics (HGSFP). C.H. acknowledges support fromthe Alexander von Humboldt foundation.
Appendix A: Calculation of the probe fieldtransmission with a non-constant atomic densitydistribution
For a realistic experimental situation we consider aGaussian atomic density distribution. However, we as- sume that the density is approximately constant in x, y -direction resulting in a distribution of the form ρ ( z ) = ρ e − z / (2 σ z ) , (A1)where ρ is the peak atomic density. Moreover, we as-sume that the density is approximately constant overthe range of the nonlinear susceptibility (local approx-imation). Inserting the atomic distribution given by Eq.(A1) into Eq. (20) and solving the differential equationresults in T = I ( z ) I = A ( z )1 − B ( z ) I (A2)for the transmission T of the probe field, where A ( z ) = exp (cid:26)
12 ˜ a √ πσ z (cid:20) Erf (cid:18) z √ σ z (cid:19)(cid:21)(cid:27) , (A3) B ( z ) = exp (cid:26)
12 ˜ a √ πσ z (cid:27) (A4) × (cid:90) z −∞ d ξ ˜ a exp (cid:26)
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