Behavior of a very large magneto-optical trap
BBehavior of a very large magneto-optical trap
A. Camara , R. Kaiser , and G. Labeyrie ∗ Institut Non Lin´eaire de Nice, UMR 7335 CNRS,1361 route des Lucioles, 06560 Valbonne, France
We investigate the scaling behavior of a very large magneto-optical trap (VLMOT) containing upto 1 . × Rb atoms. By varying the diameter of the trapping beams, we are able to changethe number of trapped atoms by more than 5 orders of magnitude. We then study the scaling lawsof the loading and size of the VLMOT, and analyze the shape of the density profile in this regimewhere the Coulomb-like, light-mediated repulsive interaction between atoms is expected to play animportant role. PACS numbers: 67.85.-d, 03.75.-b, 05.60.Gg, 42.25.Dd
I. INTRODUCTION
Since its first realization in 1987 [1] the magneto opti-cal trap (MOT) has been the working horse of cold atomexperiments and continues to be used in a large variety ofexperiments, such as Bose-Einstein condensates or degen-erate Fermi gazes, atomic clocks and sensors and quan-tum memories. In some of these experiments, increasingthe number of trapped atoms is an important advantage.Previous studies have shown that when increasing thenumber of atoms loaded into a MOT, the peak atomicdensity tends to saturate and the size of the atomic cloudincreases [2]. This has been a strong limitation to thestraightforward use of the MOT towards Bose-Einsteincondensation, requiring novel cooling techniques, oftenbased on conservative trapping potentials combined toevaporation, in order to achieve the quantum degeneracyregime. At first it seemed that by increasing the numberof atoms in a MOT, a net additional compression forcecould be obtained due to a shadow effect of the large num-ber of atoms, attenuating the incident laser beams [3].However, a more refined model, taking into account theradiation pressure force of the scattered photons showedthat the size of a MOT increases with increased atomnumber [2]. This size increase has been confirmed byexperiments [2] and is due to a modified frequency spec-trum of the scattered photons when atoms are driven atlarge values of the saturation parameter. For a vanishingincident saturation parameter, the shadow effect can atbest merely compensate the repulsion force due to thescattered photons, which explains that all experimentsup to now have observed an increase in MOT size as thenumber of atoms is increased. The most commonly usedmodel proposed in ref. [2] shows that the repulsion forceis analogous to a Coulomb repulsion between particles ofsame charge, leading to a constant density of particles ina harmonic trap. Further studies [4, 5] have shown thatin contrast to most common explanations of MOTs, notonly the velocity distribution of the trapped atoms but ∗ To whom correspondence should be addressed. also their spatial distribution might require sub-Dopplermechanisms, such as Sisyphus cooling [6].With the availability of larger laser power at the rel-evant wavelengths for cooling and trapping atoms, it ispossible to trap more and more atoms in a MOT. It istherefore important to study the MOT scaling laws forlarge atom numbers, to understand e.g. how the cap-ture velocity and the number of trapped atoms can bemaximized. This may allow for adapting designs in newexperiments where atom number and trap size are impor-tant parameters to be optimized. Another aim would beto obtain experimental signatures in the multiple scat-tering regime of the MOT that could help improving ourunderstanding of this complex situation and discriminat-ing between various available models.We thus report in this paper the results of an experi-ment where the number of trapped atoms N is varied overa wide range (more than 5 orders of magnitude) in a well-controlled way. The paper is organized as follows. A firstsection is devoted to the description of the experimentalscheme. Details of the experimental procedure are im-portant since it is known to affect the observed scalinglaws [7]. We then report in section III our measurementof the scaling law for the number of trapped atoms versusthe size of the trapping beams, which is found to increasewith an exponent larger than previously reported in theliterature. We discuss this result using a simple numeri-cal simulation based on the standard Doppler model forthe MOT. We then analyze in Sec. IV the scaling lawfor the size of the MOT as a function of the number ofatoms, and compare it to various models. Finally, wediscuss in Sec. IV the evolution of the shape and elliptic-ity of the atomic density distribution as the number ofatoms is varied. II. EXPERIMENTAL SETUP
The six independent VLMOT trapping beams are de-rived from a single beam using a 1 × a r X i v : . [ phy s i c s . a t o m - ph ] O c t FIG. 1: (Color on line) Experimental scheme. We show theMOT trapping scheme along one of the three spatial dimen-sions, corresponding to the axis of the coils generating themagnetic field gradient ( x ). The arrangement is identical forthe other two dimensions (except for the magnetic coils). ACCD is used to image the fluorescence of the cold cloud inthe ( x, y ) plane (see text for details). ter single-pass amplification through a 2W tapered am-plifier. The output fibers tips are placed in the objectfocal plane of six 10 cm-diameter, 30 cm-focal lengthlenses to obtain large (waist 2.6 cm) collimated trap-ping beams (see Fig. 1). For each dimension of spacethe corresponding pair of beams is aligned in a counter-propagating fashion. The total trapping power sent tothe atoms is 329 mW, corresponding to a peak inten-sity I = 5 mW/cm per beam. We trap Rb usinga trapping light detuned by a quantity δ MOT from the F = 2 → F (cid:48) = 3 transition. In the present paper,we will use δ MOT = -3, -4 or -5 Γ (where the naturalwidth Γ = 2 π × .
06 MHz). We use these rather largedetuning values because they maximize the number oftrapped atoms, and also to avoid the dynamical instabil-ity that arises at large numbers of atoms and smaller de-tunings [8]. In our setup, the repumping light is producedby another DFB laser tuned close to the F = 1 → F (cid:48) = 2transition. The repumper beam is superimposed to thetrapping beam (the repumper power representing a fewpercent of the total) before the injection into the taperedamplifier. Thus, the repumping light is present in each ofthe 6 VLMOT beams with the same circular polarizationas the trapping light, yielding a very symmetrical config-uration. The main remaining source of asymmetry inour setup is the slight imbalance between the intensitiesof the 6 beams, due to the specifications of the fiber-splitter. This imbalance is at most 10% for two beamsin the same counter-propagating pair. A magnetic fieldgradient of 7.4 G/cm along the axis of the anti-helmoltzcoils is applied to spatially trap the atoms.This experiment aims at measuring scaling laws for theVLMOT as the number N of trapped atoms is varied. Wetune N via the diameter of the trapping beams, usingsix large diaphragms whose aperture D is adjusted (seeFig. 1). Since the capture range of the MOT dependsstrongly [9] on D , this is an efficient and well-controlledway of varying N without changing the MOT parameters at the location of the trapped atoms.Fluorescence images are recorded in a plane containingthe magnetic gradient coils axis x , where the gradient istwice that along the two other axes. We thus have accessto the intrinsic anisotropy of the MOT shape, which isstudied in section V. To acquire the fluorescence images,we switch the trapping light detuning to δ im = −
8Γ for ashort duration of 230 µ s. This is short enough to neglectthe displacement of the atoms during the image acquisi-tion ( ≈ µ m). The large detuning employed to recordthe fluorescence images has two important consequences:first, the cloud’s optical density (OD) at the illuminatinglight’s detuning is then (cid:28) − , which allows one to ne-glect inelastic scattering and thus the resonant compo-nent of the Mollow triplet [10]). As will be discussedin section V, multiple scattering can strongly distort therecorded fluorescence intensity profiles (see Fig. 7). Sec-ond, because of the large detuning we can safely neglectthe Zeeman shift due to the magnetic field gradient whichis still on during the measurement (the maximal Zeemanshift across the MOT size is ≈ . z with a weak probe beam (waist1.5 mm). For the highest MOT beam diameter D = 94mm and a detuning δ MOT = − × assuming an equal distribution of the atomic populationamong Zeeman sub-states. III. VLMOT LOADING
Fig. 2 shows the measured evolution of the number oftrapped atoms when the beams diameter is varied (log-log scale). As can be seen, N first increases very stronglywith D : N ∝ D . . This exponent α = 5 .
82 is sig-nificantly larger that predicted by the standard model( α = 4) [9, 11]. This Doppler model, based on the bal-ance between loading rate and losses due to collisionswith background atoms, leads to: N ∝ D σ ( v c u ) (1)where σ is the collisional cross-section with backgroundatoms, v c is the velocity capture range of the MOT and u = (cid:113) k B Tm the most probable velocity in the Maxwell-Boltzmann distribution ( u = 240 m/s in our case). If one FIG. 2: (Color on line) Loading the VLMOT (experiment).We plot the number of trapped atoms N as a function ofthe diameter D of the trapping beams (see text), for threedifferent MOT detunings: δ MOT = −
3Γ (stars); δ MOT = − δ MOT = −
5Γ (squares). We observe a fast increase,followed by a progressive saturation. The line is a fit N ∝ D . of the data for δ MOT = − D , for the detuningsof Fig. 2. The lines emphasize the two observed regimes: v c ∝ D . (dotted line) and v c ∝ D . (solid line). assumes a constant force (i.e. a constant photon scatter-ing rate) acting on an atom inside the MOT volume, onefinds [11] v c ∝ √ D and from eq. 1 the scaling N ∝ D follows (assuming a v c -independent σ ).However, the assumption of a constant scattering rateduring the trajectory of an atom entering the trappingvolume is in general not verified. As the atom movestoward the trap center and is being decelerated, it grad-ually gets tuned out of resonance with the MOT laserbeams and the scattering rate decreases. An accurate es- FIG. 4: Variation of VLMOT size with N . We show twoexamples of fluorescence images (see text for details), re-spectively at low (N = 4.2 × , A ) and large (N = 1.3 × , B ) number of trapped atoms. The MOT detuningis δ MOT = − timation of the capture velocity and of its scaling with D thus requires a numerical simulation of the atomic tra-jectories. We performed such a 3D numerical simulationbased on the Doppler model, and found two regimes forthe scaling of v c with D (see Fig. 3): below a certain crit-ical value of D , which depends on both δ MOT and ∇ B , v c is roughly proportional to D (dotted line), while forlarger values of D the increase of v c is slower (solid line).We stress that this cross-over is not due to the finitewaist (2.6 cm) of the MOT beams. Instead, it is due tothe nonlinear dependency of the MOT force as a functionof velocity. For small D the capture velocity is small, andlies in the linear range of the force kv c < | δ MOT | . In thisregime, increasing D will result in an increase of the cap-ture velocity by roughly the same amount, since the forcewill increase proportionally to v c . For large D such that kv c ≈ | δ MOT | , the force is already maximal. Therefore,an increase of D will result in a much smaller increase of v c than in the linear regime.Inserting v c ∝ D into Eq. 1, we obtain N ∝ D which is in good agreement with what we measure inFig. 2 for D <
30 mm ( N ∝ D . ). The saturation ofthe number of trapped atoms at larger D is mainly dueto the cross-over seen in Fig. 3, although one expects theGaussian profile of the MOT beams to enhance this satu-ration for D >> w . Comparing Figs. 2 and 3, we observea quite striking qualitative agreement for the behavior ofthe different detunings. Finally, we note that even highernumber of atoms could be loaded in the VLMOT usinglarger beams and larger detunings, which requires higherlaser powers.
IV. VLMOT SIZE SCALING
It is known since the 90s [2] that atoms in a MOT are ingeneral not independent, but interact through exchangeof photons. The reabsorbtion of scattered photons indeed
FIG. 5: (Color on line) VLMOT size scaling. We measurethe FWHM size of the cloud along the magnetic coils axis L x as a function of the number of atoms. The three sets ofdata correspond to different MOT detunings: δ MOT = − δ MOT = −
4Γ (dots); δ MOT = −
5Γ (squares). A fitof the δ MOT = −
4Γ data for
N > × yields L x ∝ N . (solid line). The dashed line corresponds to the prediction ofthe standard model [2] L ∝ N / . generates a repulsive inter-atomic force, which tends toexpand the cloud. As a result, the size L of the cloudincreases with N , while it is independent of N in thenon-interacting, small- N regime where it is determinedonly by the MOT parameters and the temperature (hencethe name of ”temperature-limited” regime).Fig. 4 illustrates the large variation of MOT size ob-served in our situation as the number of atoms is tuned.The size varies by a factor ≈
35 while N varies by a factor31 000. To be more quantitative, we plot on Fig. 5 themeasured cloud size L x along the magnetic coils axis, asa function of N and for the three detuning values. Thissize is determined as the full width at half maximum(FWHM) of a cut of the image, through its center, alongx. Since the cloud shape is generally not Gaussian (seenext section), we do not integrate the image along y. For δ MOT = − N > × is wellfitted by L ∝ N . . This exponent is observed over alarge range of 4 decades. Similar scalings are found forthe two other detunings: L ∝ N . and L ∝ N . for δ MOT = −
5Γ and δ MOT = −
3Γ respectively. The sizesalong the weak confinement axis y , not shown in Fig. 5,are larger (see section V) and exhibit similar exponents: L ∝ N . , N . and N . for δ MOT = − , − −
5Γ respectively. For
N < , the expansion of thecloud with N seems to slow down. This is indeed ex-pected in the limit of small N where light-induced in-teractions vanish and the MOT size becomes indepen-dent of N . However, this regime is expected to occur formuch smaller atom numbers: in ref. [2], the temperature- FIG. 6: (Color on line) VLMOT peak density and opticaldensity. We plot in A the peak spatial density and in B theon-resonance optical density of the cloud obtained from thedata of Fig. 5 (see text). limited regime was observed for N < ≈ µ m for small N could be due to a poor res-olution of our imaging system. The ultimate resolution(limited by the pixel size) is of 23 µ m. Another factor lim-iting the resolution is the motion of the atoms during theimage exposure. The typical displacement correspondingto our temperature is of the order of 30 µ m. The residualeffect of multiple scattering is minimized by our choiceof a large detuning for the imaging (see fig. 7), but itsexact magnitude remains difficult to estimate. However,its impact is expected to be very small in the small N limit, where the cloud’s OD is small (see Fig. 6 B ).In the standard Doppler model of the MOT [2], theMOT size is determined by the balance between the ex-ternal trapping force, the inter-atomic repulsion, and a“shadow” compressive force due to the attenuation of theMOT beams inside the cloud [3]. The last two are “col-lective” forces which vanish in the temperature-limitedregime. Under the assumptions of ref. [2], which amountto linearizing the trapping and shadow forces and as-suming a spatially-independent Coulomb-like interactionforce, this balance yields a constant spatial density insidethe cloud: n max = cκ Iσ L ( σ R − σ L ) (2)where c is the speed of light, κ is the spring constant char-acterizing the restoring force for a single-atom MOT, σ L is the absorption cross-section for a laser photon and σ R the cross-section for the absorption of a scattered pho-ton. σ L and σ R are different due to the fact that boththe spectral and polarization properties of the scatteredlight differ from that of the laser light. In this model,increasing N thus result in an expansion of the MOT atconstant density: L ∝ N / . A good agreement withthis prediction was reported by the authors of ref. [2]for N < × , while they observed a faster increasefor larger atom numbers. Possible explanations for thisbehavior included the effect of the magnetic field gradi-ent and high-order multiple scattering of light inside thecloud. A more involved (numerical) model [7] surpris-ingly led to the same L ∝ N / scaling, although thecalculated density profiles were no longer homogeneousbut displayed a truncated Gaussian shape [7]. This modeltakes into account the nonlinear form of both trappingand shadow forces and the spatial dependence of the in-teraction force. However the interaction force takes intoaccount only double scattering (a single re-absorptionevent), as in the standard model. In ref. [7], we haveshown that using different techniques to vary the num-ber of atoms (i.e. tuning the intensity or the diameter ofa repumping beam) could yield different scaling laws forthe MOT size. This still unexplained observation hintsat the complexity of the trapping process which is intrin-sically multi-level in nature. In the experiment where thediameter of the repumping beam was used as a mean tovary N , which is closest in principle from that describedin the present paper, a scaling L ∝ N . was observedwhich is consistent with the standard model. Our presentobservation L ∝ N . is not too far off the N / predic-tion. The complex behavior of the observed MOT shapes,discussed in the next section, may be responsible for thisdeviation.Finally, we plot in Fig. 6 A the peak spatial densityof the cloud versus N for the three MOT detunings ofFig. 5. This density is inferred from the measured num-ber of atoms N and sizes L x and L y , assuming an axially-symmetric MOT L z = L y and a Gaussian density distri-bution. We find densities around a mean value of 2 × cm − , which are rather independent of N (typical varia-tion of a factor of 3 over more than 4 orders of magnitudeof variation of N ). The lowest variation of density cor-responds to the largest detuning δ MOT = − B shows the on-resonanceoptical density calculated using the same assumptions.The OD is seen to increase continuously with N with arough scaling OD ∝ N . for δ MOT = −
5Γ (as deter-mined by a fit over the whole N range) and a maximalvalue of 185 for our parameters. V. VLMOT SHAPE
In this section we discuss the evolution of the shapeof the cloud as N is varied. Indeed, as emphasized inref. [7], the density profile of the cloud may be the ul-timate signature to discriminate between various modelsrather than the L ( N ) scaling. We start by reviewing theexisting models in the various MOT regimes, as well asthe published observations.In the limit of small N (temperature-limited regime)where photon re-absorption can be safely neglected, thecloud’s density distribution is Gaussian and independentof N . For larger atom numbers, when re-absorption setsin, the standard model [2] predicts a uniform density pro-file. This results from the combination of the trapping,“shadow” and repulsive forces, with the model of ref. [2]assuming a linear spatial dependence of both the first twocompression terms and a spatially-independent repulsiveforce. It has been shown in ref. [7, 13] that includingthe full spatial dependence of these forces in the Dopplermodel yields density profiles that are truncated Gaus-sians. The size σ dens of these Gaussians is only deter-mined by MOT parameters, while the truncation radius R tr depends on the number of atoms. In the limit of small N ( R tr (cid:28) σ MOT ) one recovers a uniform density profilehas predicted by the standard model. On the contrary, inthe limit of very large N this spatially-dependent modelpredicts a Gaussian shape for the density profile.These predictions rely on the Doppler model of theMOT. However, it was realized very early after the ad-vent of the MOT that sub-Doppler mechanisms play a de-terminant role in the force near the center of the trap [6].This picture, initially developed in the framework of inde-pendent atoms, somewhat survives in the regime of multi-ple scattering albeit with a modified friction and diffusionrate leading to higher temperatures [14]. When the num-ber of atoms is further increased, the position-dependentprofile of the restoring force leads to a “two-component”regime for the MOT [4, 15]. There, a central part with ahigher density of atoms is subjected to a highly restoringsub-Doppler force, and is surrounded by a halo of lowerdensity where the force is essentially Doppler-like. Theradius R of the surface separating these two volumes isgiven by the equality of Zeeman shift and light shift ofthe ground state [4]: R ≈ (cid:126) Ω µ B ∇ Bδ MOT (3)
FIG. 7: (Color on line) Impact of multiple scattering duringimaging. We compare the fluorescence profiles obtained forthe same cloud ( N = 2 × ) but with different detuningvalues used for the imaging: δ im = −
2Γ (1), δ im = −
4Γ (2), δ im = −
6Γ (3), δ im = −
8Γ (4) and δ im = −
10Γ (5). Theinset shows the evolution of the measured FWHM with δ im . If the radius of the cloud is larger than R , the MOTis in the 2-component regime. For ∇ B = 20 G/cm, δ MOT = − I = I sat and with the parameters ofCesium, the authors of ref. [4] find that this occurs for N ≈ (with R ≈ µ m). Eq. 3 shows that formoderate N where the MOT size is not very large, thetwo-component regime may be reached for high magneticgradients and light detunings, and low Rabi frequencies.We now turn to the reported measurements of densitydistributions in a MOT. We first stress that all these wereperformed by direct fluorescence imaging of the MOT(i.e. using the actual MOT detuning for the imaging),which may cause significant distortions of the profiles asshown in Fig. 7. Here we compare the profiles obtainedfor the same cloud, but with different values of the de-tuning δ im during the imaging . The profiles obtainedclose to resonance are broader with a flatter top thanfor a detuned illumination, where the profiles become al-most independent of δ im and converge toward the atomicdensity distribution. The choice of the detuning also sig-nificantly affects the measured width, as illustrated inthe insert. We thus conclude that when the shape mea-surement is performed by direct imaging of the MOTfluorescence (and not with a detuned excitation as donehere), one should be cautious with the interpretation ofthe recorded profiles as long as the OD at the MOT de-tuning is not (cid:28) δ im = − N < × . For larger N , the standard model predicts aconstant density which integrated once yields a profile f ( x ) ∝ √ R − x where R is the radius of the uni-form sphere of atoms. Such rather flat profiles werealso observed in ref. [12], but not in a subsequent de-tailed study [4] where Gaussian profiles were observed in-stead. The “constant density” signature of the multiple-scattering regime was then observed on the peak density,similarly to what we show in Fig. 6 A . Deviations from aGaussian were also reported in ref. [17], and well fitted tothe functional dependence introduced by the authors ofref. [19] to account for multiple scattering and finite tem-perature. The difference between all these experimentalfindings is not elucidated, but we note that flat-top pro-files can also be due to multiple scattering of the illumi-nating light even if the density distribution is Gaussian,as discussed before. The two-component regime was ob-served in ref. [4, 18], and the sub-Doppler component wasnicely separated from the Doppler halo in ref. [5].We now discuss our observations. We show in Fig. 8some fluorescence profiles recorded for different atomnumbers (panels A to D ) at δ MOT = − x and y . The symbols cor-respond to the data, the lines to Gaussian profiles. Thevertical and horizontal scales are normalized to ease thecomparison. The horizontal scaling is different in pan-els A to D , and is chosen such that the FWHM of theprofiles is equal to 1. The vertical scale is logarithmic toallow for a better observation of the wings, and the scal-ing is such that the peak value of the profiles is 1. For N below typically 10 atoms, we obtain profiles quite closeto Gaussians (Fig. 8 A ). When N is increased to roughly10 (Fig. 8 B ), the profiles deviate from a Gaussian andget quite close to the flat-top shapes of ref. [17, 19]. Thisis accompanied by a steepening of the wings of the pro-files, which is a prediction of all models including mul-tiple scattering [2, 7, 19]. When N is increased evenfurther, the profile along y gradually rounds off, whilethe profile along x develops for N > a central fea-ture with enhanced density (Fig. 8 C and D ). This lastbehavior is best seen in Fig. 8 E where we plot the dataof D along x in linear scale (the arrows point at the in-flexion points in the profile). We stress that this generalbehavior is, apart from minor details in the shapes, ro-bust against modifications of the MOT alignment suchas e.g. the beam intensity imbalance. We find that allprofiles along x for N between 10 and 10 are con-sistent with a double-component distribution, includinga narrower part near the MOT center. We believe thatfor this range of atom numbers, our MOT operates inthe two-component regime. Indeed, eq. 3 yields in ourcase R ≈ ∇ B = 7 . δ MOT = − / Γ = 0 . N ≈ × ,which is in rough agreement with the appearance of the FIG. 8: Fluorescence profiles of the cloud. We plot the fluo-rescence profiles along x (dots) and y (circles) for four differentatom numbers: ( A ) N = 3 . × ; ( B ) N = 1 . × ; ( C ) N = 2 × ; ( D ) N = 1 . × . The lines correspond to aGaussian shape. The data of panels ( A ) to ( D ) are all scaledin the same way: all profiles are vertically normalized to amaximum value = 1 (note the log scale), while the horizon-tal scaling is different for all four plots such that the profile’sFWHM is equal to 1. Panel ( E ) shows the data of ( D ) along x , in linear scale. central feature. This is also in rough agreement with anextrapolation of the MOT “phase diagram” computed inref. [4]. A measurement of the velocity distribution ofthe atoms, not performed in this work, could possiblycorroborate this hypothesis. We also find that for a fixed N , the deviation from a Gaussian profile is larger whenthe detuning is smaller (MOT operating closer to reso-nance), which is to be expected for multiple scatteringeffects.The double-component behavior does not show upclearly in the profiles along y . Indeed, one expects fromeq. 3 that the radius of the sub-Doppler central fea-ture is inversely proportional to the magnetic field gradi- FIG. 9: (Color on line) Ellipticity of the cloud. We plotthe ellipticity (cid:15) of the MOT measured versus the number ofatoms N . (cid:15) is measured at 90%(half-filled circles), 50%(dots)and 10%(circles) of the peak value of the fluorescence images.The shaded area corresponds to the limits obtained from themodel of ref. [20], while the dashed line (cid:15) = √ ent. We thus expect for the central feature an ellipticity (cid:15) = L y /L x = 2. However, the ellipticity of the Dopplercomponent which makes up most of the cloud’s size isonly of the order of 1.5 as can be seen on Fig. 9. Thewidths of the sub-Doppler and Doppler components arethus more similar along y , rendering their differentiationdifficult. Fig. 9 shows the cloud’s ellipticity measuredversus N , at different proportions of the peak value inthe fluorescence images: 90% (stars), 50% (dots) and10% (squares). It can be seen that the ellipticities mea-sured at 10 and 50% of the maximum are following aquite parallel evolution when N is varied, with valuesaround 1.5 for N > . On the contrary, the ellipticitymeasured at 90% of the maximum (i.e. near the cen-ter of the cloud) show a steep increase for N > andreaches higher values at large N (average of (cid:15) = 2 . N > ). This behavior is consistent with the appear-ance of a two-component distribution for high N values.The shaded area on Fig. 9 corresponds to the possiblevalues of (cid:15) according to the model of ref. [20]. The au-thors of this recent work proposed the measurement ofthe ellipticity as a mean to determine experimentally thecross-section ratio σ R σ L (see eq. 2), an interesting quantitydifficult to compute in a realistic MOT situation. Theirmodel rely on the standard approach of ref. [2], using thesame hypothesis (small OD, double scattering only, andspatially-independent cross-sections σ L and σ R ). It pre-dicts a variation of (cid:15) with the MOT parameters (intensityand detuning), but not with N as it is observed in thepresent work (Fig. 9). This is not surprising, however,since we expect these assumtions to break down at large N values. Furthermore, our complicated MOT shape be-havior is clearly not accounted for by this model. FIG. 10: (Color on line) Ellipticity of the cloud versus MOTdetuning. We plot here the ellipticity measured at 50% of thepeak value of the fluorescence images, for different values of δ MOT : δ MOT = −
3Γ (stars); δ MOT = −
4Γ (dots); δ MOT = −
5Γ (squares). The shaded area corresponds to the limitsobtained from the model of ref. [20], while the dashed line (cid:15) = √ Fig. 10 shows how (cid:15) (measured at 50% of the peak fluo-rescence) depends on δ MOT . We observe globally that forintermediate atom number 10 < N < × , (cid:15) increaseswith | δ MOT | . In the framework of ref. [20] this would cor-respond to a strong increase of σ R σ L (note however that for δ MOT = −
5Γ our measured (cid:15) largely exceeds the theo-retical limit of 1.81). Interestingly, for
N > × allcurves collapse together and seem to converge towardsthe temperature-dependent limit (dashed line). This cor-responds roughly to the situation where the cloud’s opti-cal density at δ MOT becomes larger than 1. In this regimewhich is clearly beyond the reach of the standard modelof ref. [2], the trapping laser beams are strongly attenu-ated inside the cloud. A more refined model needs to be developed to understand how shadow effect and multiplescattering concur to yield the observed behavior.
VI. CONCLUSION
In this paper we have presented our observations onthe behavior of a very large magneto-optical trap con-taining up to 1 . × atoms. To our knowledge, thisis the largest number of atoms in a MOT reported in theliterature. The number of trapped atoms and the cloud’ssize and shape are studied as a function of the diameter D of the MOT’s lasers beams. Using this technique, theatom number can be varied by 5 orders of magnitude.We observe an increase of N with D much faster thanpreviously reported, a feature well-reproduced by simu-lations of the MOT’s capture velocity based on a simpleDoppler model. We find a scaling of the cloud size ver-sus N roughly consistent with the standard model of aMOT in the multiple scattering regime, even up to suchlarge numbers of atoms. A careful measurement of thecloud shape yields Gaussian profiles up to 10 atoms,and then strong deviations for larger N . For N > ,our observations are consistent with the two-componentregime for the MOT, in agreement with the predictionsof ref. [4]. Such large MOTs where strong multiple scat-tering effects constitute interesting tools to search foranalogies with e.g. plasma physics, hydrodynamics, orstellar physics [8]. They can also be used to producelarge (centimeter-scale) cold clouds with a high opticaldensity, well-suited to perform original experiments ine.g. nonlinear optics [21] and self-organization [22]. Acknowledgments
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