Signature of collective effects in the frequency-comb-induced radiation pressure force
Mateo Kruljac, Danijel Buhin, Domagoj Kovacic, Vjekoslav Vulic, Damir Aumiler, Ticijana Ban
SSignature of collective effects in the frequency-comb-induced radiation pressure force
M. Kruljac, D. Buhin, D. Kovaˇci´c, V. Vuli´c, D. Aumiler, and T. Ban Institute of Physics, Bijeniˇcka cesta 46, 10000 Zagreb, Croatia (Dated: January 21, 2021)We investigate the modifications of the frequency-comb-induced radiation pressure force on cold Rb atoms that are induced by the collective effects. Collective effects include both coherent andincoherent contributions and depend on the optical thickness of the atomic cloud. We observereduction and broadening of the comb-induced force when the cloud’s optical thickness increases,and compare the measured results with predictions of a coherent model based on the timed Dickestate approach and an incoherent scattering model based on the shadow effect explained by Beer-Lambert law. Both models describe the experimental results well, indicating that an incoherentscattering approach is sufficient to explain the observed modifications of the comb-induced forceeven for larger optical thicknesses. The results support the analogy between the frequency comband continuous-wave laser-atom interaction and thus pave the way toward novel frequency combapplications in laser cooling, quantum communication, and light-atom interfaces based on structuredand disordered atomic systems.
PACS numbers: 37.10.De, 37.10.Vz
I. INTRODUCTION
Optical frequency combs (FCs) have become an un-avoidable source of light in applications ranging frommetrology [1, 2] and high-resolution spectroscopy [3–5] toprecision ranging [6] and calibration of atomic spectro-graphs [7]. In recent years, the applications of FCs haveexpanded to laser cooling and trapping of atoms and ions,and quantum communication. Recent demonstrations in-clude FC cooling of ions [8, 9], neutral atoms [10, 11], andsimultaneous dual-species FC cooling [12]. Applicationsof the FC for quantum communication are based on therecent progress in generating highly multimode nonclas-sical FCs [13–16] and the potential for the realization ofmulti-mode quantum memories [17, 18]. For these noveland intriguing applications of FCs that are at the coreof quantum technologies, it is necessary to understandthe FC light-matter interactions, and to verify whetherand to what extent these interactions are comparable toalready known continuous-wave (cw) laser-matter inter-actions. In this respect, it is particularly important tounderstand FC light scattering and all effects that ac-company it.Collective effects observed in light scattered from anensemble of cold atoms illuminated by a cw laser havebeen an extremely fruitful platform for studying light-matter interactions [19–23]. The light scattered by theatomic ensemble contains coherent and incoherent con-tributions. The coherent contributions result from theconstructive interference of the light scattered by theatomic emmiters, and provides the basis of optical co-operative effects [23–25]. These contributions include co-herent backscattering and Mie scattering [26], and canbe explained using a coherent dipole model [27, 28], or,in the case of cooperative scattering in the low-intensityregime, by the use of timed Dicke state approach [25].The incoherent contribution is incoherent in a sense thatthe phase of the light scattered from different emitters is random. This contribution includes diffuse scattering[26], and can be explained by the random walk model[27, 28]. Diffuse scattering induces the progressive at-tenuation of the laser intensity as it propagates throughthe cloud, which is called the shadow effect and can beexplained by the Beer-Lambert law [26]. In addition,anisotropy of the emission pattern is observed, but thiscontribution is much smaller than the contribution of theshadow effect.Over the past years, several experiments studied theradiation pressure force exerted by a cw laser on a coldatomic cloud [25, 28–30], to clarify the interplay betweencollective coherent and incoherent contributions to theradiation pressure force. Bienaime et al. [29] developeda cooperative radiation pressure force model based onthe timed Dicke state approach and compared it withthe experiment. Based on the results, they proposed touse the radiation pressure force as a new tool for the ex-perimental investigation of cooperativity. This idea wasextremely attractive and led to a series of new studies.Those studies, however, later indicated that specific ef-fects observed in radiation pressure force may not alwaysbe a signature of cooperativity, i.e. coherent collectiveeffects, but a result of different mechanisms associatedwith incoherent collective effects [26, 31].In this paper we report on the role of collective ef-fects on the FC-induced radiation pressure force actingon a cold cloud of Rb atoms released from a magneto-optical trap (MOT). We have measured the FC force asa function of the comb mode detuning from the Rb | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition for differentoptical thicknesses of the atomic cloud. Reduction andbroadening of the FC-induced force are observed as thecloud’s optical thickness is increased. In order to under-stand the role of coherent and incoherent contributionsto the FC-induced radiation pressure force, experimen-tal results are compared to the predictions of the coher-ent timed Dicke state approach and an incoherent model a r X i v : . [ phy s i c s . a t o m - ph ] J a n based on the Beer-Lambert law. Both models satisfacto-rily describe the experimental results, suggesting that theincoherent scattering approach is sufficient to explain theobserved modifications of the FC force even in the case oflarger optical thicknesses. Our results support the con-siderations in [26] for the case of cw-induced force andsmall optical thicknesses, thus verifying the analogy be-tween FC and cw light-atom interactions.The paper is organized as follows. In Sec. II we de-scribe in detail the experimental setup, including thepreparation of cold cloud of Rb atoms of a given opti-cal thickness and its determination, as well as the proce-dures for measuring the FC radiation pressure force. InSec. IIIA, we present the measurements of the FC radi-ation pressure force as a function of comb detuning, andshow that two distinct peaks appear in one f rep scan, re-flecting the interaction of three comb modes with threehyperfine transitions. The FC force is measured for dif-ferent atom densities. In Sec. IIIB, we adopted the the-oretical models developed in the literature for the case ofcw radiation pressure force and applied them to calculatethe FC-induced force. Given the very low intensity of thecomb mode involved in the FC-atom interaction, we useda coherent model based on timed Dicke state approachand an incoherent model based on the Beer-Lambert lawinvolving the shadow effect. In Sec. IIIC, we presentthe measured and calculated results of the reduction andbroadening of the FC-induced force as a function of thecloud optical thickness, showing the agreement of bothmodels with the measurements. We finish in Sec. IVwith conclusion and outlook. II. EXPERIMENT
A simplified scheme of the experimental setup for thepreparation of a cold Rb cloud and its characterizationusing absorption imaging, as well as the setup for FCradiation pressure force measurement using fluorescenceimaging is shown in the Fig. 1(a).
Preparation of a cloud of cold atoms . A cold Rbcloud is loaded from a background vapour in a stainlesssteel chamber using a standard six-beam configuration.The preparation of a cold Rb cloud of a given opticalthickness is achieved in three consecutive stages: MOTloading, temporal dark MOT, and repumping stage. Inthe first stage, we load the MOT for 6 s, with the coolinglaser detuned − .
5Γ from the Rb | S / ; F = 2 (cid:105) →| P / ; F (cid:48) = 3 (cid:105) transition, and the repumper laser inresonance with the | S / ; F = 1 (cid:105) → | P / ; F (cid:48) = 2 (cid:105) transition, generating a cloud of ≈ · atoms at atemperature of around 50 µ K, and a 1 /e radius of ≈ .
07 MHz is the natural linewidthof Rb | S / (cid:105) → | P / (cid:105) hyperfine transition [32]. Inthe second stage, we apply a 15 ms long temporal darkMOT, where we reduce the power of the repumper laserto 10 µ W and the detuning of the cooling laser to − on-resonanceoptical thickness b b P (mW) repumper n ( × c m - ) peak density n fluorescenceimagingFC FORCE AOM
FC beam absorption imaging
OPTICAL THICKNESS absorption imaging beam xy z
PBS M
MOT beams
MOT beams
MOT beams MOT beams (a)(b)
FIG. 1: (a) A simplified experimental scheme. Two pairs ofcounter-propagating MOT beams are shown, while the thirdpair is propagating in the z axis. The absorption imagingbeam and the FC beam are co-propagated in the x -axis. Theoptical thickness is measured using the absorption imagingcamera, while the FC force is measured using the fluores-cence imaging camera. During the measurement of opticalthickness, the FC beam is blocked using an AOM. M is a mir-ror and PBS is a polarizing cube. (b) On-resonance opticalthickness (red circles) and cloud peak density (green trian-gles) as a function of the repumper laser power during thefirst, MOT loading stage. Solid lines represent a guide to theeye. atoms are pumped into the F = 1 ground state, whichcauses an increase of the cloud density and, consequently,of the optical thickness. Finally, we increase the powerof the repumper laser to 1.5 mW and tune the coolinglaser to − b ( y, z ), using the standard off-resonant ab-sorption imaging technique. We image the spatial in-tensity distribution of a weak probe beam propagatingin the x direction with no cloud present, I ( y, z ), andwith the beam passing through a Rb cloud, I ( y, z ),and calculate the optical thickness given by b ( y, z ) = − ln [ I ( y, z ) /I ( y, z )]. For an atomic cloud of a Gaus-sian density distribution, b ( y, z ) will also have a Gaus-sian shape. By fitting a 2D Gaussian to the mea-sured b ( y, z ), we extract the optical thickness at thecentre of the cloud, b peak , as well as R y , R z cloudradii. From b peak we calculate the on-resonance opticalthickness, b , using b = b peak · (cid:0) δ img / Γ (cid:1) , where δ img is the detuning of the probe laser frequency usedfor absorption imaging from the relevant atomic reso-nance frequency. In our case, the probe beam is 2.5cm in diameter, has 30 µ W of power and is 9.1 MHzred detuned from the | S / ; F = 2 (cid:105) → | P / ; F (cid:48) =3 (cid:105) transition, giving b = 10 b peak . The imagingpulse duration is 50 µ s. On-resonance optical thick-ness is defined as b = σ (cid:82) ∞−∞ n ( x, y = z =0) dx , where σ is on-resonance cross section [32], and n ( x, y, z ) = n exp (cid:2) − x / (2 R x ) − y / (2 R y ) − z / (2 R z ) (cid:3) is the Gaus-sian spatial density of the cloud with n equal to peakdensity in its centre. The number of cold atoms in thecloud, N , and peak cloud density, n , are then calcu-lated from the measured optical thickness and radii using N = (2 πb R x R y ) /σ and n = 3 N/ (4 πR x R y R z ).In order to vary the optical thickness of the cloud, wechange the power of the repumper laser in the first MOTloading stage, leaving the dark MOT and the repumpingstage parameters unchanged. By changing the repumperpower in the 5 µ W - 1.5 mW range, the optical thick-ness measured for the | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition can be varied from 1 to 21. However, changingthe power of the repumper laser in the first stage alsoaffects the other cloud parameters such as size, numberof atoms, density, and temperature. Although a changein these parameters does not affect the accuracy of opti-cal thickness determination since it is measured directlyby absorption imaging, a detailed characterization of allcloud parameters is made as a function of the repumperlaser in the first stage. In Fig. 1(b), the peak density andon-resonance optical thickness are shown as a functionof the repumper laser power in the first, MOT loadingstage. Both values initially increase linearly and satu-rate at higher repumper powers. The largest achievedon-resonance optical thickness is 21, the peak density is1.3 · cm − which gives n k − = 2 · − , indicating acondition in which multiple scattering effects can be ne-glected [27, 31]. Here, k =2 π/ (780 nm) is the wave vectorrelevant for the excitation of the Rb | S / (cid:105) → | P / (cid:105) transition [32]. For the given range of repumper laserpowers, the cloud temperature varies from 35 µ K to 75 µ K, where the temperature is measured using a standardtime-of-flight (TOF) technique.
FC force measurement . The FC is generated byfrequency doubling an Er:fiber mode-locked femtosecondlaser (TOPTICA FFS) operating at 1560 nm with a rep-etition rate of f rep =80.495 MHz. The frequency-doubledspectrum used in the experiment is centered around 780nm with a FWHM of about 5 nm and a total outputpower of 76 mW. The FC spectrum consists of a series ofsharp lines, i.e. comb modes [33]. The optical frequencyof the n -th comb mode is given by f n = n · f rep + f ,where f rep is the laser repetition rate and f is the offsetfrequency. In our experiment, we actively stabilize f rep and f n by giving feedback to the cavity length and pumppower of the mode-locked laser, thus indirectly fixing f .The detuning of the n -th comb mode with respect tothe | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition is variedby scanning f while keeping f rep fixed. A detailed de-scription of the FC stabilization and scanning scheme ispresented in our recent papers [11, 12].After the cold cloud is prepared, it is illuminated by theFC beam, exerting a FC-induced radiation pressure forceon the cloud’s centre of mass (CM). The force is measuredas a function of the detuning of the n -th comb mode withrespect to the | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) tran-sition. The experimental setup and the measurementsequence used are similar to the ones described in ourrecent works [11, 12]. A linearly polarized FC beam issent through an acousto-optic modulator (AOM) for fastswitching, and is directed to the center of the cloud. Thetotal power of the FC beam on the atoms is 25 mWand the beam size (FWHM) is 2.7 mm, resulting in thepower and intensity per comb mode of about 0.75 µ Wand 9 µ W/cm , respectively. The measurement sequencestarts with the preparation of a cloud of a given opticalthickness. At t = 0 we turn off the MOT cooling beamsand switch on the FC beam. The MOT repumper lasersare left on continuously to optically pump the atoms outof the | S / ; F = 1 (cid:105) ground level. They are arranged ina counter-propagating configuration with the intensitypredominantly in the direction perpendicular to the FCbeam propagation, and have no measurable mechanicaleffect. The quadrupole magnetic field is also left on. Welet the comb interact with the cold cloud for 0.5 ms. Dur-ing this time the center of mass of the cloud acceleratesin the FC beam direction (+ x -direction) due to the FCradiation pressure force. The FC beam and repumperlasers are then switched off, and the cloud expands freelyfor a variable time t free , after which we switch on theMOT cooling beams for 0.15 ms and image the cloud’sfluorescence with a camera. The times t free are chosento maximize both the CM displacement and the signal-to-noise ratio and are varied from 4 ms (for the smallest b ) to 18 ms (for the largest b ). This measurement se-quence is repeated 3 times for each detuning of the FCmode from the | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transi-tion, and the resulting fluorescence images are averaged.The cloud’s CM displacement in the + x -direction is de-termined from the images, providing information on thecloud’s acceleration and the FC radiation pressure force.We then change the optical thickness of the cloud andrepeat the measurement sequence.It is worth noting here that the approaches to chang-ing the optical thickness of the cloud by changing the re-pumper laser power immediately after dark MOT stageused in [29], and by changing the cloud’s free expan-sion time used in [19] are not applicable in our case ofthe FC excitation. In the first approach, only a frac-tion of atoms are transferred from | S / ; F = 1 (cid:105) to | S / ; F = 2 (cid:105) ground level after the dark MOT, depend-ing on the repumper laser power. Atoms remaining in the | S / ; F = 1 (cid:105) level and atoms in | S / ; F = 2 (cid:105) could besimultaneously excited by different comb modes, whichwould result in a complex lineshape of the measured FCforce. In the second approach, the size of the FC beamshould be at least twice the initial size of the cloud, whichcannot be achieved in our setup due to the low power percomb mode. III. RESULTS AND DISCUSSIONA. FC force as a function of cloud density
In Fig. 2(a) we show the measured FC force as afunction of the FC detuning δ , which we define as thedetuning of the n -th comb mode from the | S / ; F =2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition, for different peak clouddensities, n . Due to the nature of the comb spectrum,the FC radiation pressure force is periodic with respectto the comb detuning with period equal to f rep . Twodistinct peaks appear in one f rep scan reflecting the in-teraction with three comb modes, as explained in detailin our recent work [11]. The peak at δ = 0 is due to the n -th comb mode being in resonance with the | S / ; F =2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition, whereas the peak at δ ≈ − . n − | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 2 (cid:105) transition, and the ( n − | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 1 (cid:105) transition. Forcompleteness, in Fig. 2(b) we show the calculated FCforce, obtained by summing the contributions from threehyperfine transitions (for details on FC force calculationssee [11]).As the cloud density increases, reduction and broad-ening of both FC force peaks is observed. In addition,the ratio of the peaks at δ = 0 and δ ≈ − . n approaches zero, as it reflectsthe ratio of the | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) and | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 2 (cid:105) transition dipole matrixelements [32]. This behaviour can be easily understood,given the well-known finding that collective effects suchas force reduction and broadening depend on the opti-cal thickness rather than the density [27]. The observeddependence of the peak ratio on the cloud density is theresult of the different optical thicknesses of the two peaks for a given optical density. As the optical thickness is de-fined through the cross section σ = hω Γ / (2 I sat ), where I sat is the saturation intensity depending on the dipolemoment of the relevant transition [32], the two peakshave different optical thicknesses for a given density andtherefore different factors of force reduction, which di-rectly affects the peak ratio. In the following sectionswe will therefore present and analyze the dependence ofthe FC force on the optical thickness for each force peakseparately. -40 -30 -20 -10 0 10 20 (b) cm -3 cm -3
12 ×10 cm -3 F N F C ( a r b . un it s ) (a) F F C ( a r b . un it s ) δ (MHz) p ea k r a ti o n / 10 cm -3 F = = = = = = FIG. 2: (a) Measured FC force as a function of the FCdetuning δ , for different peak atom densities n . The in-set shows the ratio of the FC peak forces at δ = 0 and δ ≈ − . δ . The total FC force (violet line)is obtained by summing the force contributions from three | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 1 , , (cid:105) hyperfine transitions. B. Theoretical models
In this work we use an analytical expression for thecw cooperative radiation force in the presence of disor-der developed in [29], based on the timed Dicke state,as well as the results of an incoherent model based onthe Beer-Lambert law to calculate the FC force. Wecompare both theoretical approaches with experimentalresults and draw a conclusion about the role of incoher-ent and coherent scattering in the FC radiation pressureforce. In doing so, we imply that only a single comb mode(the one closest to resonance) dominates the light-atominteraction, i.e. the effects associated with other combmodes can be neglected since they are detuned from theatomic resonances [11, 12]. In this respect we can con-sider a single comb mode participating in the interactionas a cw laser.The radiation pressure force acting on the j -th atom,ˆ F j , can be derived from the laser-atom interaction Hamil-tonian describing the excitation of N two-level atoms witha cw laser beam propagating in ˆ z direction, giving:ˆ F j = ˆ F aj + ˆ F ej , (1)where ˆ F aj is the force contribution associated with ab-sorption of photons from the laser beam and ˆ F ej is theforce contribution arising from emission of photons in ar-bitrary directions. The average radiation pressure forcedefined as ˆ F = (1 /N ) (cid:80) j ˆ F j = ( F tot /N )ˆ z accelerates thecloud’s CM in the direction of beam propagation, withthe acceleration given by ˆ a CM = ˆ F /m where m is thesingle atom mass. A detailed derivation of the averagecw radiation pressure force resulting from the excitationof N atoms by a resonant laser can be found in [34], andis given by: F = hk Γ4 πN (cid:90) π dφ (cid:90) π dθ sin θ (1 − cos θ ) I s ( θ, φ ) . (2) I s ( θ, φ ) is the scattered far-field intensity given by I s ( θ, φ ) = (cid:80) Nj =1 | β j | + (cid:80) Nj (cid:54) = m =1 ( β j β ∗ m exp[ − i k · ( r j − r m )]), where the angles θ, φ determine the direction of theemitted photon with the wave vector k , and β j and r j are the atomic coherence and position of the j -th atom,respectively. Equation (2) shows that the radiation pres-sure force that accelerates the atoms along the directionof the incident laser beam is proportional to the net ra-diation flux of the scattered intensity, i.e. the pattern ofthe scattered intensity is directly mapped to the radiationpressure force. In addition, it should be noted that theforce has two contributions: one incoherent that is pro-portional to excitation probability of individual atomicdipoles (cid:80) Nj =1 | β j | , and the other coherent due to theinterference between the different atomic dipoles.In the case of incoherent scattering, the second termin I s ( θ, φ ) is equal to zero, and the light is scat-tered isotropically. I s can be calculated from thetransmitted intensity governed by the Beer-Lambertlaw, i.e. I s = I [1 − exp( − b ( x, y )], where I isthe intensity of the incident laser beam. b ( x, y ) = b / (cid:0) δ / Γ (cid:1) · exp (cid:2) − x / (2 R x ) − y / (2 R y ) (cid:3) , with b equal to on-resonance optical thickness and δ to the de-tuning of the incident laser frequency from the atomicresonance frequency. Due to the Beer-Lambert law, thelaser intensity attenuates as it propagates through thecloud, and the radiation force decreases accordingly. Thiseffect is called the shadow effect. The radiation pressureforce in the presence of the shadow effect is derived from Eq. (2) in [26], and is given by: F shadow F = Ein( b ) b , (3)where Ein( b ) is the entire function given by Ein( z ) = (cid:82) z dx (1 − e − x ) /x , with b = b / (cid:0) δ / Γ (cid:1) , and F thesingle-atom radiation pressure force. F has a Lorentzianshape and for small intensities of the incident laser, I /I sat (cid:28)
1, is given by F = σ I / (cid:2) ck (cid:0) δ / Γ (cid:1)(cid:3) .The inclusion of coherent effects in the force calcu-lations requires the second term in I s ( θ, φ ). A conve-nient model to calculate this full, many-body problemis to use the coupled dipole model and to expand theatomic coherences β j for n -th order scattering events[26, 27, 35, 36]. This approach is beyond the scope ofthis work. Instead, a mean-field approach inspired bythe timed Dicke state model is used. This model as-sumes that all atoms are driven by the unperturbed laserbeam, i.e. the atoms acquire the phase of the laser andall have the same excitation probabilities characterizedby β j = β/ √ N · exp( i k · r j ). Timed Dicke state approachhas become widely used in recent years, as it gives a de-scription of experimental results on superradiance, sub-radiance and frequency shifts that are observed in lightscattered by cold atomic ensembles. Such emission iscalled cooperative because it is the result of cooperativeactivity of many coherent atomic dipoles [24], and the ra-diation pressure force that results from such an emissionis called cooperative radiation force. cw cooperative ra-diation force is studied in detail in [29, 30]. The averageforce, F coop , can be estimated by: F coop F = 4 δ + Γ δ + (1 + b / Γ (cid:20) b k R ) (cid:21) , (4)where R = R x + R y + R z is the cloud radius, and F is the single atom force given above. The model neglectsreabsorption of photons by other atoms, and is usuallyused in the limit of small intensities or large detunings.In this work we apply equations (3) and (4) to ourexperimental parameters, and calculate the average radi-ation pressure force on the CM of a cold Rb cloud inthe presence of shadow (i.e. incoherent collective), andcooperative (i.e. coherent collective) effects and compareit to the measured FC force induced by a single FC combmode.
C. Collective effects in the FC force
In Fig. 3 we show the measured FC force as a func-tion of the FC detuning δ , for different on-resonance op-tical thicknesses b , in the case of | S / ; F = 2 (cid:105) →| P / ; F (cid:48) = 2 , (cid:105) excitations. In the case of the | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition, b is mea-sured directly as described in Sec. II, and divided by2.8 to obtain b relevant for the | S / ; F = 2 (cid:105) →| P / ; F (cid:48) = 2 (cid:105) transition. The same figure also showsthe cooperative radiation pressure force calculated for ourexperimental parameters using Eq. (4), where δ is the de-tuning of the n -th comb mode from the | S / ; F = 2 (cid:105) →| P / ; F (cid:48) = 3 (cid:105) transition, Γ = 6 .
07 MHz, k = 2 π/ (780nm), R =0.3 mm and b corresponds to the measured val-ues. -10 -5 0 5 10 150.00.20.40.60.81.0 b = = = = F N F C ( δ ) / F F C ( δ = ) δ (MHz) F =
2 F' = (a) -5 0 5 10 15 b = = = = F N F C ( δ ) / F F C ( δ = - . M H z ) δ (MHz) F =
2 F' = (b) b = = = = FIG. 3: Measured FC force (symbols) and cooperative ra-diation pressure force calculated using Eq. (4) (lines) as afunction of δ , for different optical thicknesses b . (a) FC forceis due to the n -th comb mode being in resonance with the | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition. (b) FC forceis due to the ( n − | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 2 (cid:105) transition. The measured FC forces arising from the | S / ; F =2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition show a Lorentzian lineshape in the whole range of measured b , for 0 . ≤ b ≤ .
8, Fig. 3(a). In the case of the | S / ; F = 2 (cid:105) →| P / ; F (cid:48) = 2 (cid:105) transition, the FC forces deviate from theLorentzian line shape, Fig. 3(b), due to the | S / ; F =2 (cid:105) → | P / ; F (cid:48) = 1 (cid:105) FC force contribution positioned inthe blue wing of the peak, as indicated in the Fig. 2(b). For a given b , the Lorentzian function is fitted to theexperimental data to obtain the FC force line center,linewidth, peak value, and offset. While the FC forceoffset should be zero, experimentally we see a small off-set due to inaccuracies in determination of the initial andfinal position of the cloud’s CM, from which the acceler-ation, i.e. the force is determined. The small FC forceoffset is subtracted from all experimental data shown inthe Figs. 3 and 4.The FC force reduction and broadening is clearly ob-served for both peaks shown in Figs. 3(a) and 3(b) andpresented in more details in Fig. 4. The reduction is de-fined as F NF C ( δ ) /F F C ( δ ), and shown in Fig. 4(a) for δ = 0and δ = − Γ in the case of | S / ; F = 2 (cid:105) → | P / ; F (cid:48) =3 (cid:105) transition, and for δ = − . δ = − . − Γ in the case of | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 2 (cid:105) transition. In order to obtain F F C ( δ ) and to normalizeour data, the measured FC force values as a function of b for a given detuning δ , F NF C ( δ ), are fitted with theEq. 4 with F F C ( δ ) as a free fitting parameter. Thusdetermined F F C ( δ ) is then used as a scaling factor fornormalization of the FC forces shown in Figs. 2, 3 and4(a).In Fig. 4(a) we show the measured FC force reductionalong with the predictions of theoretical models discussedabove, i.e. shadow and cooperative effects, calculated us-ing Eqs. (3) and (4), respectively. We observe a reductionof the FC force with increasing b . The FC force reduc-tion is larger when the relevant comb mode is resonantwith a given atomic transition, i.e. the n -th comb modein resonance with the | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition ( δ = 0), and the ( n − | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 2 (cid:105) transition ( δ = − . b and support the experimental results.In Fig. 4(b) we show the measured FC force linewidthsΓ F C as a function of b along with predictions of the dis-cussed theoretical models of shadow and cooperative ef-fects, calculated using Eqs. (3) and (4), respectively. Fora given b , Γ F C is obtained from the fit of a Lorentzianfunction to the FC force experimental data. We ob-serve the increase of the FC linewidth with increasing b . At b = 0, the FC linewidth of Γ = 6 .
07 MHzis expected, as it reflects the natural linewidth of the Rb | S / (cid:105) → | P / (cid:105) transition [32]. For the largest b = 20 . . b (violet solidline in Fig. 4(b)) as this model does not include multi-ple scattering effects that can induce the flattening of theforce linewidth curve at large b [30]. The force linewidthin the presence of the shadow effect calculated using Eq.(3) recovers the linear dependence of the force linewidthfor small b and predicts a nonlinear dependence of thelinewidth for a large b (green dashed line in Fig. 4(b)).Although for the measured range of b the difference in = = δ = - Γ F = = δ = -25.5 MHz- Γ F = = δ = = = δ = -25.5 MHz F N F C ( δ ) / F F C ( δ ) (a) Γ F C ( M H z ) b F = = F = = (b) FIG. 4: Signature of collective effects in the FC force arisingfrom the Rb | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 2 (cid:105) and Rb | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transitions. (a) MeasuredFC force reduction (symbols) and calculated force reductionin the presence of shadow (dash line) and cooperative (solidline) effects, using Eq. (3) and (4), respectively, as a functionof b for δ = 0 and δ = − Γ in the case of | S / ; F = 2 (cid:105) →| P / ; F (cid:48) = 3 (cid:105) transition, and δ = − . δ = − . − Γ in the case of | S / ; F = 2 (cid:105) → | P / ; F (cid:48) = 3 (cid:105) transition. (b) Measured FC force broadening (symbols) andcalculated force broadening in the presence of shadow (greendashed line) and cooperative (violet solid line) effects, usingEq. (3) and (4), respectively, as a function of b . the force linewidths predicted by these two models is notsignificant, in Fig. 4(b) there are indications that theshadow effect model more accurately supports our ex-perimental results. IV. CONCLUSION
In conclusion, we have measured the frequency-comb-induced radiation pressure force acting on a cold Rb cloud in the presence of collective effects which dependon the cloud’s optical thickness. We observed reductionand broadening of the frequency comb force as the op-tical thickness increases. Theoretical models for cw ra-diation pressure force in the presence of shadow and co-operative effects developed in [26] and [29], respectively,are used to describe the measured frequency comb force.Both theoretical models support the experimental resultswell, thus verifying the analogy between the comb andcw atom-light interaction, which is in line with previ-ous investigations of the comb-induced force on atoms[11, 12]. In addition, showing a good agreement of bothtimed Dicke state and shadow effect models with mea-sured comb-induced force, we have extended the validityof the conclusions presented in [26] even for larger opticalthicknesses.Our results show that the influence of off-resonancecomb modes on the comb-atom interaction is minor andcan be neglected, even in the case of increased opticalthickness of the cloud. In addition, our results indicatethat the observed frequency comb force reduction andbroadening arise as a result of the shadow effect describ-ing the progressive attenuation of the light intensity inthe cloud due to the Beer-Lambert law and that it istherefore not necessary to use numerically more demand-ing coherent atom-light interaction models to describethe observed effects. The results presented in this pa-per contribute to the understanding of scattering of fre-quency comb light by an ensemble of cold atoms, whichpave the way toward novel frequency comb applicationsin the field of cooling and trapping, quantum communi-cation, and light-atom interfaces based on structured anddisordered atomic systems.
V. ACKNOWLEDGEMENT
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