Vibrational mechanics in an optical lattice: controlling transport via potential renormalization
A. Wickenbrock, P. C. Holz, N. A. Abdul Wahab, P. Phoonthong, D. Cubero, F. Renzoni
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Vibrational mechanics in an optical lattice: controlling transport via potentialrenormalization – Supplemental Material
A. Wickenbrock, P.C. Holz, N.A. Abdul Wahab, P. Phoonthong, D. Cubero, and F. Renzoni
I. MODEL AND DEFINITIONS
As a model for our experiment, we consider the sim-plest configuration in which Sisyphus cooling has beenshown to take place, i.e. the case of a J g = 1 / → J e =3 / ⊥ lin config-uration [1]. In Ref. [2], the following generalized Fokker-Planck equation was found in the semiclassical limit forthe probability density P ± ( z, p, t ) of each atom that is inthe ground state sublevel |±i = | J g = 1 / , M g = ± / i at the position z with momentum p : (cid:20) ∂∂t + pm ∂∂z − U ′± ( z ) ∂∂p + F ( t ) ∂∂p (cid:21) P ± = − γ ± ( z ) P ± + γ ∓ ( z ) P ∓ + ∂ ∂p [ D ± ( z ) P ± + L ± ( z ) P ∓ ] , (1)where m is the atomic mass and U ′± ( z ) = dU ± ( z ) /dz ; U ± ( z ) = U − ± cos(2 kz )] (2)is the optical bipotential created by the laser fields, with k the laser field wave vector; F ( t ) is a time-dependentdriving force that can be generated by phase modulatingone the lattice beams [3]; γ ± ( z ) = Γ ′ ± cos(2 kz )] (3)is the transition rate between the ground state sublevels,with Γ ′ the photon scattering rate; D ± ( z ) = 7¯ h k Γ ′
90 [5 ± cos(2 kz )] (4)is a noise strength coefficient describing the random mo-mentum jumps that result from the interaction with thephotons without transition between ground state sub-levels; and L ± ( z ) = ¯ h k Γ ′
90 [6 ∓ cos(2 kz )] (5)is related to random momentum jumps that appear influoresecence cycles when the atom undergoes a transi-tion between the atomic sublevels. The normalizationcondition is given by Z d z Z d p [ P − ( z, p, t ) + P + ( z, p, t )] = 1 . (6) We consider two different types of time-dependentdriving forces: a bi-harmonic drive of the form F d ( t ) = A cos( ωt ) + A cos(2 ωt + φ ) , (7)and a high-frequency (HF) drive of the form F HF ( t ) = A HF sin( ω HF t + φ ) . (8)The bi-harmonic drive is used to probe the potentialamplitude. The HF drive determines the potential renor-malization, as discussed in next Section. II. POTENTIAL RENORMALIZATION BYHIGH-FREQUENCY DRIVING
We now study the effect on the cold atom system of ahigh-frequency signal F HF , of the form of Eq. 8 with φ an arbitrary initial phase. A low-frequency bi-harmonicdrive F d ( t ) is also included in the analysis, to model theexperiments in which the potential is probed by usingthis type of driving.We are interested in situations in which the frequency ω HF is much larger than ω and any other characteristicfrequency in the system. Formally, this can be achievedby taking the asymptotic limit ω HF → ∞ . In this limit, itis also necessary that A HF → ∞ if the HF signal is to haveany effect. Due to this strong driving, the momentumchanges very rapidly, since its time-derivative is of order A HF . Integrating this dominant term in time, we find arapidly changing contribution to the position z ( t ) thatgoes as − r sin( ω HF t + φ ), where r = A HF mω . (9)Formally, we will consider the asymptotic limit ω HF , A HF → ∞ while keeping r fixed. By extractingthe fast dependence from z ( t ),ˆ z ( t ) = z ( t ) + r sin( ω HF t + φ ) , (10)it is expected that ˆ z ( t ) changes on a much slower time-scale than that of the HF signal. The density probabili-ties for the new variable are then given byˆ P ± (ˆ z, ˆ p, t ) = P ± [ˆ z − r sin( ω HF t + φ ) , ˆ p − rmω HF cos( ω HF t + φ ) , t ] , (11)where ˆ p = m d ˆ z/dt . The corresponding generalizedFokker-Planck equations are (cid:20) ∂∂t + ˆ pm ∂∂ ˆ z − ˆ U ′± (ˆ z, t ) ∂∂ ˆ p + F d ( t ) ∂∂ ˆ p (cid:21) ˆ P ± = − ˆ γ ± (ˆ z, t ) ˆ P ± + ˆ γ ∓ (ˆ z, t ) ˆ P ∓ + ∂ ∂ ˆ p h ˆ D ± (ˆ z, t ) ˆ P ± + ˆ L ± (ˆ z, t ) ˆ P ∓ i , (12)where ˆ U ′± (ˆ z, t ) = U ′± [ˆ z − r sin( ω HF t + φ )], ˆ γ ± (ˆ z, t ) = γ ± [ˆ z − r sin( ω HF t + φ )], and similarly for ˆ D ± (ˆ z, t ) andˆ L ± (ˆ z, t ). These coefficients depend on time only throughthe HF signal. On the other hand, both ˆ P ± and F d varywith time on a much longer timescale. Therefore, wecould remove the time dependence from the above coef-ficients by integrating over a time interval that includesmany HF periods but in which ˆ P ± and F d do not appre-ciably change. Equivalently, we can eliminate that fastdependence by noting that ˆ P ± should be independent ofthe HF phase φ . Integrating Eq. (12) over the phase φ , we finally find a generalized Fokker-Planck equation analogous to (12) but with the following coefficients:¯ U ± (ˆ z ) = 12 π Z π d φ U ± [ˆ z − r sin( ω HF t + φ )]= U − ± J (2 kr ) cos(2 k ˆ z )] , (13)¯ γ ± (ˆ z ) = Γ ′ ± J (2 kr ) cos(2 k ˆ z )] , (14)¯ D ± (ˆ z ) = 7¯ h k Γ ′
90 [5 ± J (2 kr ) cos(2 k ˆ z )] , (15)¯ L ± (ˆ z ) = ¯ h k Γ ′
90 [6 ∓ J (2 kr ) cos(2 k ˆ z )] , (16)where J (2 kr ) = 12 π Z π d φ cos(2 kr sin φ ) (17)is the Bessel function of the first kind. Eqs. (13)–(16)describe the system renormalization by the HF field inthe asymptotic limit ω HF → ∞ . Equivalently, it canalso be seen as the lowest order of a multiple time-scaleformalism using the expansion parameter ε = ω/ω HF (seefor example [4]). [1] G. Grynberg and C. Mennerat-Robilliard, Phys. Rep. ,335 (2001).[2] K. Petsas, G. Grynberg, and J.-Y. Courtois, Eur. Phys. J.D , 29 (1999). [3] F. Renzoni, Adv. At. Mol. Opt. Phys. , 1 (2009).[4] J. Casado-Pascual, Chem. Phys. , 170 (2010). r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Vibrational mechanics in an optical lattice: controlling transport via potentialrenormalization
A. Wickenbrock , P.C. Holz , N.A. Abdul Wahab , P. Phoonthong , D. Cubero , and F. Renzoni Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom and Departamento de F´ısica Aplicada I, EUP, Universidad de Sevilla,Calle Virgen de ´Africa 7, 41011 Sevilla, Spain and F´ısica Te´orica,Universidad de Sevilla, Apartado de Correos 1065, Sevilla 41080, Spain (Dated: November 10, 2018)We demonstrate theoretically and experimentally the phenomenon of vibrational resonance in aperiodic potential, using cold atoms in an optical lattice as a model system. A high-frequency (HF)drive, with frequency much larger than any characteristic frequency of the system, is applied byphase-modulating one of the lattice beams. We show that the HF drive leads to the renormalizationof the potential. We used transport measurements as a probe of the potential renormalization.The very same experiments also demonstrate that transport can be controlled by the HF drive viapotential renormalization.
PACS numbers: 05.40.-a, 05.45.-a, 05.60.-k
The control of transport is a recurrent topic in physics,chemistry and biology. The typical scenario correspondsto particles diffusing on a periodic substrate, with trans-port controlled by the application of dc and ac externalfields [1, 2]. The ultimate limit for the control of trans-port is often the impossibility of tuning the periodic po-tential, as it is usually the case in solid state.In this work we provide a proof-of-principle of howthis limitation can be overcome, and demonstrate the-oretically and experimentally the control of a periodicpotential amplitude via a strong high-frequency (HF) os-cillating field. The potential is renormalized, with its am-plitude controlled by the strength and frequency of theHF field. The mechanism underlying the potential renor-malization is the so-called vibrational resonance, intiallyintroduced [3] and observed [4–7] in bistable systems.Our experiment uses cold atoms in a dissipative opti-cal lattice as a model system. However, the phenomenondemonstrated here is very general, and is relevant to anyclassical system of particles in a periodic potential. Thismay also offer a possibility of tuning the potential in solidstate systems, where this is usually considered impossi-ble. Combined with previous work which showed howac fields can be used to control transport via dynamicalsymmetry breaking [1, 2] and tunnel coupling renormal-ization [8, 9], the present work demonstrates that a com-plete control of transport can be achieved via ac fields.Our experimental work relies on the study of the trans-port properties of atoms in an optical lattice for differentstrengths of the applied HF field. We will demonstratethat by tuning the HF field it is possible to control theamplitude of the potential, and to make it vanish. Inthis respect, the use of cold atoms in dissipative opti-cal lattices is very convenient as the transport propertiesin these systems have been studied in detail [10–12], andthis allows us to use the transport measurements to char-acterize the potential. We will provide two different setsof measurements, as supporting evidence of the poten- tial renormalization. First, we will demonstrate that thediffusion properties, which are known to strongly dependon the potential depth [10, 11], can be controlled by theHF field in a way which corresponds to the potentialrenormalization. Second, we will show that also directedtransport, as induced by harmonic-mixing (HM) [13] of abi-harmonic drive, can be controlled by the HF field. Infact, anharmonicity, together with the breaking of a dy-namical symmetry, leads to the creation of directed cur-rents in harmonic mixing. Therefore, whenever the po-tential, renormalized by the HF field, vanishes, directedtransport should cease [14–16].Before discussing the experimental results, we intro-duce a model useful for the understanding of the po-tential renormalization of a dissipative optical latticeas produced by a HF oscillating field. We considerthe simplest model of a dissipative optical lattice: a J g = 1 / → J e = 3 / m , illuminatedby two counterpropagating laser fields with orthogonallinear polarizations. This configuration generates a 1Doptical lattice [12]. The atom in the ± ground state ex-periences the potential U ± ( z ) = U [ − ± cos(2 kz )] / z is the laser beam propagation axis, k the laserfield wavevector and U the optical lattice depth [12, 17].We now introduce a HF oscillating force with frequency ω HF and amplitude A HF : F HF ( t ) = A HF sin( ω HF t + φ ) , (1)with φ a (mainly irrelevant) phase which describes thestate of the oscillating force at t = 0. Of interest here isthe high-frequency case, where the frequency of the HFdrive is much larger then any characteristic frequency ofthe system, in the present case the vibrational frequency ω v of the atoms at the bottom of the well. In the asymp-totic limit of infinite amplitude and frequency of the drive( ω HF → ∞ , A HF → ∞ ), it is possible to show [18] that,consistently with Refs. [3, 14–16], the atomic dynamicscorresponds to the motion in a static (i.e. without HFfield) dissipative optical lattice, with renormalized am-plitude ˜ U :˜ U ± (ˆ z ) = U [ − ± J (2 kr ) cos(2 k ˆ z )] / J is the Bessel function of the first kind, and r = A HF / ( mω ) is the parameter –here and thereaftertermed the HF ratio – which controls the renormalizationof the optical lattice.The above analysis shows that, in the asymptotic limitof infinite frequency and strength, a HF field leads to aneffective renormalization of the potential. We now con-sider finite values of driving away from infinity that areexperimentally accessible. The atomic transport in theoptical lattice in the presence of a HF field is numericallystudied for two different set-ups, which correspond to theones used to provide the experimental evidence.In the first set-up, a HF force of finite amplitude andfrequency is applied to atoms in a dissipative optical lat-tice. For this scheme, the effective renormalization ofthe optical potential can be detected by studying the dif-fusion properties of the atoms through the lattice. Infact, for a dissipative optical lattice of the type consid-ered here, it is well established [10, 11] that there is acritical potential depth located at about U cr ∼ E r ,with E r = ~ k / (2 m ) the recoil energy, which sepa-rates two very different regimes. For potential depthslarger than the critical one, the diffusion is normal. In-stead, for potential depth lower than the critical one,the diffusion becomes anomalous, with the exponent ofthe diffusion dependent on the potential amplitude. Tobe quantitative, we define the diffusion exponent α as h x ( t ) i − h x ( t ) i ∼ t α in the limit t → ∞ . Accordingto this definition, α = 1 corresponds to normal diffusion,while α > α increasing for decreasing potential depth. Thuswe will take the exponent of the diffusion as a measure ofthe potential depth. To assess the effective renormaliza-tion of the potential by a HF field, we numerically simu-lated the dynamics of the atoms in a deep optical latticewith a HF drive. We determined the diffusion exponentas a function of the HF ratio r which for infinite fre-quency and amplitude of the drive determines the latticerenormalization. Our results, reported in Fig. 1, showthat the diffusion exponent α can be controlled by theHF field, with a dependence consistent with the poten-tial renormalization derived in the infinite limit, Eq. (2): α increases whenever the potential depth is decreased,with the largest values of α produced by the r valuescorresponding to the zeros of the Bessel function, i.e. tovanishing potentials. The upper bound of α = 3 for U = 0 corresponds to the vanishing of the friction mech-anism ( Sisyphus cooling [12]) associated with the opticallattice. Figure 1 also reports the value of U which cor-responds to the exponent α for an undriven lattice, so tomake explicit the correspondence between a driving withHF ratio r and the depth of the renormalized potential. Finally, we notice that our results for the diffusion expo-nent α essentially coincides with the values derived in theinfinite limit. We can thus conclude that the potentialis effectively renormalised according to the dependenceobtained in the infinite limit (see Eq. (2)). kr α Γ ’=10 ω r Γ ’=5 ω r kr U / E r : FIG. 1. Left axis: numerical results, as obtained by MonteCarlo simulations, for the spatial diffusion exponent as a func-tion of the HF ratio r for an optical lattice with a depth U = 200 E r . Right axis: value of U which corresponds tothe exponent α for an undriven lattice. Triangles correspondto results obtained in the infinite limit, for a photon scatteringrate Γ ′ = 5 ω r and Γ ′ = 10 ω r , were ω r is the recoil frequency.The filled diamonds refer to simulations with a HF field of fi-nite amplitude, with frequency ω HF /ω v = 20 and Γ ′ = 10 ω r .The solid line is a guide for the eye for the results correspond-ing to the infinite limit with Γ ′ = 10 ω r . The dotted line is | J (2 kr ) | . The error bars on the numerical results correspondto the finite statistics of the Monte Carlo simulations. In the second set-up, besides the HF drive, a bi-harmonic force of the form F ( t ) = F [ A cos( ωt ) + A cos(2 ωt + φ )] (3)is also applied to the atoms in the lattice. Here ω isthe frequency of the drive, of the same order of mag-nitude or smaller than the vibrational frequency, and φ the relative phase between harmonics. The amplitude ofthe lattice, and its renormalization by the HF field, canbe determined by observing the directed motion of theatoms through the lattice. In fact, the two harmonics ofthe drive are mixed by the nonharmonic potential, thusproducing directed motion of the atoms through the lat-tice [19]. The average current being proportional to thenonharmonicity of the potential, directed transport mea-surements give access to the potential amplitude. Moreprecisely, for weak driving the average atomic velocity isexpected to be of the form v = v max sin( φ − φ d ) , with φ d a dissipation-induced phase lag [19]. In our simulationswe determined the velocity v for different values of thephase φ , so to derive the maximum velocity v max . Thenby varying the strength of the HF drive, we were able todetermine v max as a function of the r -parameter, with re-sults as in Fig. 2. Once again, the results produced witha field of large, but finite, frequency and amplitude essen-tially coincide with those obtained in the infinite limit.These results also show that current measurements canbe used to probe the potential renormalization. When-ever the HF field leads to a shallower potential, as fromEq. (2), the current is reduced, with zero current ob-served for those values of r leading to a vanishing poten-tial. kr v m a x / v r FIG. 2. Numerical results, as obtained by Monte Carlo sim-ulations, for the amplitude of the current v max , rescaled bythe recoil velocity v r , as a function of the HF ratio r un-der a biharmonic driving force of the form of Eq. (3) with A = A = 1, F = 140 ~ kω r and ω = ω v . The solid linecorresponds to the results obtained in the infinite limit andthe diamonds to the simulation results with ω HF /ω v = 20,both cases with Γ ′ = 10 ω r . The dotted line is | J (2 kr ) | . Theerror bars on the numerical results correspond to the finitestatistics of the Monte Carlo simulations. Our experimental demonstration of potential renor-malization via HF field relies on the two detectionschemes outlined above. In both set-ups, Rb atoms arecooled and trapped in a magneto-optical trap (MOT).After a compression phase of 50 ms, and 8 ms of opti-cal molasses, the atoms are loaded into a 1D dissipativeoptical lattice. The lattice is created by the interfer-ence of two linearly polarized and counter-propagatinglaser beams, red detuned from resonance with the D -line F g = 2 → F g = 3 atomic transition. One of the lat-tice beams is sent through a double pass electro-opticalmodulator (EOM), so to be able to apply a HF phase-modulation. In the reference frame of the lattice, such aphase modulation translates into a rocking force of theform of Eq. (1). Quantitatively, a phase modulation α ( t )leads to a force F ( t ) = m ¨ α ( t ) / (2 k ) in the reference frameof the lattice [19]. In the experiments, the modulation isprogressively turned on starting after 1 ms equilibrationtime in the optical lattice, with a turn-on ramp of 1 ms.Thereafter the procedure differed for the two set-ups.In the first experiment, we study the diffusion of the atoms in the optical lattice in the presence of the HFdrive. The width of the atomic cloud is measured byfluorescence imaging after diffusive expansion inside thedriven lattice. The width is measured at a fixed set of ex-pansion times within the interval between 1 ms and 16 msfrom the lattice turn-on. The time range over which im-ages are taken is limited by the atom loss, particularly im-portant for the values of the HF ratio r leading to a van-ishing renormalized potential. Measurements of the spa-tial width of the atomic cloud on such a short temporalrange do not allow us to derive an accurate value of theexponent of the diffusion α [20]. Instead, we characterizethe diffusion by an effective diffusion coefficient D , as ob-tained by fitting the data with h x ( t ) i − h x ( t ) i = 2 Dt .Clearly, superdiffusion leads to a large enhancement ofthe derived effective diffusion coefficient. Thus a largeincrease in the effective diffusion coefficient can be takenas signature of the reduction of the potential, as producedby the renormalization by the HF field. rad/s rad/s rad/s D ( r ) / D ( r = ) k r |J (2kr)| FIG. 3. Experimental results for the effective diffusion coeffi-cient D as a function of the HF ratio r for different values of ω HF , as indicated in the figure. The data are rescaled by thevalue of the diffusion constant for an undriven lattice. The vi-brational frequency of the atoms at the bottom of the well, asdetermined by measuring the lattice beam power and waist,is ω v = (9 ± · rad/s. The solid line, with values on theright axis, is | J (2 kr ) | . Our experimental results for the effective diffusion co-efficient as a function of the HF ratio r are reported inFig. 3. The data clearly show that the atomic diffusionis significantly modified by the HF drive, with a depen-dence of the effective diffusion coefficient on the HF ratio r consistent with the potential renormalization (see e.g.Eq. (2) for the analytic expression in the limit of infinitefrequency and amplitude). Indeed, the effective diffusioncoefficient increases whenever the HF ratio r correspondsto decreasing depth of the optical lattice, with the largestvalues of the diffusion constant observed in correspon-dence of the values of r leading to a vanishing (in the in-finite limit) optical lattice. This shows that the HF driverenormalizes the optical potential, in agreement with thegeneral theory [3] and with our numerical analysis for thespecific system.In the second experiment, we probe the amplitude ofthe renormalized potential by studying directed trans-port following harmonic mixing of two harmonics, as out-lined in the numerical analysis. With respect to the pre-vious experiment devoted to the study of the atomic dif-fusion, an additional bi-harmonic drive, with frequencies ω , 2 ω and phase difference φ is introduced. This is doneusing additional acousto-optical modulators (AOMs). Inthe reference frame of the lattice, the bi-harmonic phasemodulation corresponds to a driving force of the form ofEq. (3). In the experiment, the HF driving is first rampedup, as in the previous experiment. Then the biharmonicdrive is progressively turned on with a ramp-up time of4 ms. The velocity of the center-of-mass of the atomiccloud is derived by position measurements obtained viafluorescence imaging. The measurements are repeatedfor 10 different values of the phase difference φ betweenharmonics. The data are then fitted by the expected de-pendence v = v max sin( φ − φ d ), thus deriving a value for v max which can be taken as a measure of the renormal-ized potential depth. In fact, the mixing of harmonicsrequires an anharmonic potential, with the current gen-erated proportional to the anharmonicity. Our resultsfor v max as a function of the HF ratio r are presented inFig. 4. These data for the directed transport amplitudeare consistent with the renormalization of the potentialdepth by the HF drive. In fact, whenever the value ofthe HF ratio r corresponds to a reduced potential depth,the current decreases, with zero current observed for the r -values corresponding to the zeros of the Bessel func-tion, a signature of the vanishing optical lattice. Theseresults also demonstrate a new scheme for the control ofthe transport via ac fields: the amplitude of the currentcan be controlled by a variation in the HF field and thedirection reversed via a π -shift in the relative phase be-tween harmonics. Finally, we notice that there is a smalldeviation, both in the experiment and in the numericalsimulations (see Fig. 2) from the behaviour expected fromthe Bessel function at small values of r , with the datashowing an extra peak at kr ∼ .
75. This peak couldbe explained by a superimposed resonance correspond-ing to the matching of the frequency of the biharmonicforce with the oscillation frequency of the atoms at thebottom of the renormalized well.In conclusion, in this work we demonstrated experi- mentally the phenomenon of vibrational resonance in adissipative optical lattice. The application of a HF drive,with frequency much larger than any characteristic fre-quency of the system, leads to the renormalization of thepotential. The renormalized amplitude can be controlled rad/s rad/s rad/s v m a x / v r k r | J (2 kr )| FIG. 4. Experimental results for the amplitude of the current v max , rescaled by the recoil velocity v r , of directed transportthrough the optical lattice as a function of the HF ratio r .In addition to the HF drive, a biharmonic force of the formof Eq. (3) is applied to the atoms, with parameters A = 1, A = 2, ω = 9 . · rad/s, F = 112 ~ kω r . The vibrationalfrequency of the atoms at the bottom of the well is ω v =(9 ± · rad/s. The solid line, with values on the rightaxis, is | J (2 kr ) | . by the HF drive parameters. We used transport measure-ments as a probe of the potential renormalization. Thevery same experiments also demonstrated that transportcan be controlled by the HF drive via potential renor-malization.The possibility to renormalize a potential via ac fields,as demonstrated here, is very general, and it is applica-ble to any system of particles in a periodic potential. Assuch, it paves the way to the control of potentials in sys-tems in which they are not directly accessible, and it mayalso be applicable to solid state systems where ac drivescan be introduced by the application of electric fields.Finally, our set-up can also be taken as the demon-stration of a sensor able to detect signals with frequencyexceeding any internal frequency of the sensor [14–16].Here, the signal detected is the HF drive whose presence,although not coupling to any internal mode of the sys-tem, can be precisely detected due to its effect via thepotential renormalization.We acknowledge financial support from the Lever-hulme Trust, the Ministerio de Ciencia e Innovaci´on ofSpain FIS2008-02873 (DC), and the DAAD (P.C. H). [1] P. Reimann, Phys. Rep. , 57 (2002). [2] P. H¨anggi and F. Marchesoni, Rev. Mod. Phys. , 387 (2009).[3] P. Landa and P.V.E. McClintock, J. Phys. A , L433(2000).[4] V. N. Chizhevsky, E. Smeu, and G. Giacomelli, Phys.Rev. Lett. , 220602 (2003).[5] J. Casado-Pascual and J. P. Baltan´as, Phys. Rev. E ,046108 (2004).[6] J. Casado-Pascual, D. Cubero, and J. P. Baltan´as, Euro-phys. Lett. , 50004 (2007).[7] D. Cubero, J.P. Baltan´as, and J. Casado-Pascual, Phys.Rev. E , 061102 (2006).[8] A. Eckardt, C. Weiss, and M. Holthaus, Phys. Rev. Lett. , 260404 (2005).[9] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zen-esini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. ,220403 (2007).[10] S. Marksteiner, K. Ellinger, and P. Zoller, Phys. Rev. A , 3409 (1996).[11] H. Katori, S. Schlipf, and H.Walther, Phys. Rev. Lett. , 2221 (1997).[12] G. Grynberg and C. Mennerat-Robilliard, Phys. Rep. , 335 (2001).[13] K. Seeger and W. Maurer, Solid State Commun. , 603(1978); W. Wonneberger and H.J. Breymayer, Z. Phys.B: Condens. Matter , 329 (1981); F. Marchesoni, Phys.Lett. A , 221 (1986).[14] M. Borromeo and F. Marchesoni, Europhys. Lett. ,362 (2005).[15] M. Borromeo and F. Marchesoni, Phys. Rev. E ,016142 (2006).[16] M. Borromeo and F. Marchesoni, Phys. Rev. Lett. ,150605 (2007).[17] See Supplemental Material at [URL] for a complete defi-nition of the model.[18] See Supplemental Material at [URL] for the completederivation of the system renormalization by the HF field.[19] F. Renzoni, Adv. At. Mol. Opt. Phys.57