Anisotropic long-range interaction investigated with cold atoms
Vincent Mancois, Julien Barré, Chang Chi Kwong, Alain Olivetti, Pascal Viot, David Wilkowski
AAnisotropic long-range interaction investigated with cold atoms
Vincent Mancois , , Julien Barr´e , Chang Chi Kwong , Alain Olivetti , Pascal Viot , , and David Wilkowski , , MajuLab, International Joint Research Unit UMI 3654,CNRS, Universit´e Cˆote d’Azur, Sorbonne Universit´e,National University of Singapore, Nanyang Technological University, Singapore PAP, School of Physical and Mathematical Sciences,Nanyang Technological University, 637371 Singapore Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee,Sorbonne Universit´e, CNRS UMR 7600, 4, place Jussieu, 75005 Paris, France Institut Denis Poisson, Universit´e d’Orl´eans, CNRS,Universit´e de Tours, et Institut Universitaire de France Centre for Quantum Technologies, National University of Singapore, 117543 Singapore and Universit´e Cˆote d’Azur, CNRS, LJAD, 06108 Nice, France (Dated: June 29, 2020)In two dimensions, a system of self-gravitating particles collapses and forms a singularity infinite time below a critical temperature T c . We investigate experimentally a quasi two-dimensionalcloud of cold neutral atoms in interaction with two pairs of perpendicular counter-propagatingquasi-resonant laser beams, in order to look for a signature of this ideal phase transition: indeed,the radiation pressure forces exerted by the laser beams can be viewed as an anisotropic, andnon-potential, generalization of two-dimensional self-gravity. We first show that our experimentoperates in a parameter range which should be suitable to observe the collapse transition. However,the experiment unveils only a moderate compression instead of a phase transition between the twophases. A three-dimensional numerical simulation shows that both the finite small thickness of thecloud, which induces a competition between the effective gravity force and the repulsive force dueto multiple scattering, and the atomic losses due to heating in the third dimension, contribute tosmearing the transition. PACS numbers: 05.20.-y, 04.40.-b, 05.90.+m, 37.10.De, 37.10.Gh
I. INTRODUCTION
When particles interact with a force decaying at largedistance like r − α where α is less than the space dimen-sion, the force is long-range, and the system displayssome intriguing features both at and out of equilibrium[1]. However, these systems, especially those involvingattractive forces, are often not easily accessible experi-mentally.Since near resonant laser beams induce effective long-range interactions in cold atomic clouds [2, 3], it hasbeen suggested that they could be original experimentaltestbeds for long-range interactions. There are two typesof effective long-range forces. First, Dalibard [4] identi-fied the so-called ”shadow effect” in cold atomic cloudstrapped by counter-propagating laser beams. Here, ab-sorption of the near resonant laser beams, as they prop-agate inside the cloud, creates an intensity imbalancebetween the two counter-propagating beams, resultingin an effective long-range attraction between atoms. Inthe small optical depth regime, this force is similar toone dimensional (1D) gravity, i.e. the force between twoatoms does not depend on the distance. In standardthree dimensional (3D) optical molasses, there are threeorthogonal pairs of counter-propagating beams, and thecombined shadow effect looks like three 1D gravitationalinteractions directed along each pair of beams. In partic-ular, although the force is anisotropic and does not derivefrom a potential, its divergence is identical to gravity: it may then trigger a ”pseudo gravitational collapse” [4].However, a few years after the shadow effect was identi-fied, D. Sesko et al . [3] showed that multiple scattering ofphotons inside the clouds also creates an effective long-range force, but of repulsive nature, similar to a Coulombforce in the small optical depth regime. This repulsiveinteraction is typically of the same order of magnitude asthe shadow effect, but generally slightly stronger. Then,the shadow effect merely renormalizes the repulsive force,and its exotic signatures are difficult to pinpoint. As therepulsive force typically dominates, the cloud rather be-haves as a non-neutral plasma [5–7].Adding anisotropic traps, one can modify the geome-try of the cloud in order to decrease the strength of therepulsive force, and ultimately make the shadow effectdominant. For instance, Chalony et al. [8] have arguedtheoretically and demonstrated experimentally that laserinduced interactions in a thin cigar-shaped cloud bearsimilarity with 1D gravity. Similarly, Barr´e et al. [9]have suggested that the shadow effect could be dominantin a thin pancake-shaped cloud and, neglecting multiplescattering and the resulting repulsive force, argue thata collapse transition may occur if the attractive forceis strong enough. Since the divergence of the attractiveforce is identical to the gravity case, such a collapse wouldbe similar to the one happening for 2 D self-gravitatingsystems in the canonical ensemble [10], or in the Keller-Segel model of bacterial chemotaxis [11, 12], but with anon-potential force. a r X i v : . [ phy s i c s . a t o m - ph ] J un We report here on an experiment inspired by Ref. [9]:a cold atomic cloud is loaded in a very flat, pancake-shaped, optical trap, i.e. with one very stiff direction. Itis then subjected to two perpendicular pairs of counter-propagating laser beams in the easy plane of the trap.We observe a fast but moderate compression of the cloud,whereas a 2 D model predicts further compression even-tually leading to a collapse of the atomic cloud. A morerealistic description of the experiment is proposed by in-troducing a 3 D model which includes the finite thicknessof the cold atom, the associated repulsive forces due tothe multiple scattering and some atoms losses due to thefinite depth of the optical dipole trap.The paper is organized as follows. Since the shadoweffect in a pancake-shaped cloud bears similarity with2D gravity, we review this analogy in Sec. II. We firstpresent the 2 D model of self-gravitating particles, and itsphase diagram in the canonical ensemble, which shows acollapse transition below a critical temperature. We com-pare it with the 2 D model of the shadow effect, wherethe attractive force does not derive from a potential, asopposed to its true 2 D gravity counterpart. We thenidentify the parameters for which a collapse should beobserved, based on the simplified 2 D analysis. In Sec.III, we present the experimental set-up and the resultsshowing a finite compression of the cloud due to the at-tractive force, but not as strong as picted in Sec. II.We highlight the presence of atoms losses due to spon-taneous emission heating in the third direction. In Sec.IV, we attempt to bridge the gap between the simplified2 D model of Sec. II and the actual experiments of Sec.III, by considering more realistic 3D models. The firstmodel takes into account the finite thickness of the coldatomic cloud in the third dimension and the repulsiveCoulomb-like force. We numerically solve the associatedSmoluchowski-Poisson equation, and analyze how the 2 D collapse transition is smeared out by this finite thicknessof the cloud. We also propose a phenomenological exten-sion of the previous 3D model accounting for the heatingdue to spontaneous emission, which provides an estimateof the typical time to spill the atoms out of the exter-nal optical trap in the third direction, orthogonal to thepancake-shaped trap. This more realistic model qualita-tively reproduces the experimental results. In the con-clusion, we suggest some possible improvements on theexperimental set-up in order to reach larger compression. II. TWO-DIMENSIONAL MODELS:SELF-GRAVITATING SYSTEMS, CHEMOTAXISAND COLD ATOMS
In order to highlight the universality of the collapsephenomenon for systems interacting with gravitational(or quasi-gravitational) forces, we first briefly review thewell-studied self-gravitating and chemotaxis systems be-fore introducing the cold atom system for which a similarbehavior is expected. A. D self-gravitating systems For a two-dimensional system of thermalized, self-gravitating Brownian particles, the dynamics is describedby the overdamped limit of the Langevin equations (seefor instance [13])˙ r i = Gmη (cid:88) j (cid:54) = i r j − r i | r j − r i | + (cid:115) k B θmη χ i ( t ) , (1)where r i and m are the position and mass of particle i , G is the gravitational coupling, η is the viscous frictioncoefficient, θ the temperature, and k B the Boltzmannconstant. The χ i are independent Gaussian white noisessatisfying (cid:104) χ i ( t ) (cid:105) = 0 and (cid:104) χ i ( t ) χ j ( t (cid:48) ) (cid:105) = δ ij δ ( t − t (cid:48) ).We introduce dimensionless variables x i = r i /L i , s = t/τ , and ξ i ( s ) = τ / χ i ( t ≡ τ s ). Then (cid:104) ξ i ( s ) (cid:105) = 0 and (cid:104) ξ i ( s ) ξ j ( s (cid:48) ) (cid:105) = δ ij δ ( s − s (cid:48) ), and Eq. (1) becomes d x i ds = 2 π (cid:88) i (cid:54) = j x j − x i | x j − x i | + √ T ξ i ( s ) (2)with τL = 2 π ηGm , T = 2 k B θπGm . (3)Thus, the motion is controlled by a single parameter, thedimensionless temperature T . We note also that only theratio τ /L appears, so there is still some freedom in thechoice of τ and L which will be used later to rewrite theexternal harmonic trap confinement in a dimensionlessway [see Eq. (12)]. The unusual 2 /π factor is introducedto facilitate the comparison with the shadow effect in coldatomic clouds (see Sec. II C below).Associated with the Langevin description of the system[see Eq. (2)], the time evolution of the density ρ ( x , s )is governed at the mean-field level by a Smoluchowski-Poisson equation ∂ρ ∂s = ∇ · [ ρ ∇ Φ + T ∇ ρ ] , (4)where Φ is the mean gravitational potential induced bythe particles that satisfies the Poisson equation∆Φ = 4 ρ . (5)It turns out that Eq. (4) has a critical temperature T c = 1 / (2 π ) [10]. For T > T c , the system receives heatfrom the thermostat, which is transformed continuouslyin potential energy, and the system expands continuouslywithout limit [14, 15]. In a bounded domain, or in thepresence of a confining potential (absent in Eq. (4)), astable equilibrium is eventually reached. Conversely for T < T c , the heat flows from the system to the reservoir,the system shrinks and develops a singularity in finitetime. This low temperature phase can be stabilized by ashort-range repulsive interaction: the collapse transitionis then replaced by a transition with a formation of adense core [16]. B. Chemotaxis
In biology, interaction between organisms (bacteria,amoebae, cells) may be driven by chemotaxis [11, 12, 17–20]. A simple stochastic version of these models is de-scribed by a Smoluchowski equation for the density oforganisms ρ as in Eq. (4), where the mean poten-tial is replaced by (minus) a density of secreted chemical n ( r , t ). The time evolution of the chemical is given by areaction-diffusion equation. ∂n∂t = D c ∆ n − kn + λρ , (6)where D c is the diffusion constant of the chemical, k itsrate of degradation and λ its rate of production. If thedynamics of the chemical is fast with respect to the dy-namics of the density ρ , and if the degradation ratecan be neglected, Eq. (6) can be replaced by a Poissonequation similar to Eq. (5), and the dynamics of the bac-terial density is described by the same system as the 2Dself-gravitating particles. C. Cold atoms as a quasi 2D self-gravitatingsystem
We now consider a cold atom cloud confined in apancake-shaped strongly anisotropic harmonic trap, seeFig. 1. The vertical size L z is supposed to be so small thatthe system can be reduced to a quasi 2D system in the xy plane. Two orthogonal pairs of counter-propagating laserbeams, near-resonance but red detuned with respect toan atomic transition, form a 2D optical molasses in the xy plane: We shall refer to them as the long range inter-action (LRI) beams. For such a geometry, it is suggestedin [9] that the interactions inside the cloud are dominatedby the shadow effect. Indeed, the vertical dimension of-fers a route for scattered photons to escape the cloudbefore being reabsorbed. This effect is reinforced choos-ing the laser polarization within the xy plane. Thus,our model neglects multiple scattering and its associatedrepulsive force. Then the attractive shadow effect, al-beit being non conservative, bears similarities with grav-ity. The system may exhibit an extended/collapsed phasetransition at a critical temperature, similar to the ther-malized self-gravitating systems and chemotactic models.The near-resonant lasers are along the ˆ x and ˆ y axeswith the same optical frequency ω L . Their interactionwith atoms is characterized by an on-resonance satura-tion parameter s = I/I s , with I the laser intensity perbeam and I s the saturation intensity. They address aclosed two-level transition of linewidth Γ with a detuning¯ δ = ( ω L − ω ) / Γ <
0. The radiation pressure componentof the light-atom interaction for a low optical depth anda low saturation parameter can be written [8] x X y z
FIG. 1. Depiction of the beam configuration. The atomiccloud (grey) is confined in a horizontal pancake-shaped opti-cal trap, obtained with a focused elliptical far-off-resonancebeam (cyan arrow). The 2D artificial ”pseudo-gravity” is cre-ated using the four contra-propagating LRI beams (red ar-rows). The cloud is imaged with a resonant probe (blue). Allthose beams are propagating in the horizontal plane. Anotherdipole beam (green arrow), propagating along the verticalaxis, increases the cloud’s initial density. The linear polar-ization axis of each beam is indicated by a rigid bar normalto the arrows. F ( r ) = (cid:126) k Γ2 s [ (cid:40) b x ( r ) − (cid:42) b x ( r )]ˆ x + [ (cid:40) b y ( r ) − (cid:42) b y ( r )]ˆ y δ , (7)where the optical depth at position r = ( x, y, z ) seen bythe laser coming from −∞ is given by (cid:42) b x ( r ) = σ N (1 + 4¯ δ ) (cid:90) x −∞ dx (cid:48) ρ ( r (cid:48) ) , (8)with σ = πk the on-resonance absorption cross-section, k the wavenumber, and N the atom number. (cid:40) b x ( r ),corresponding to the contra-propagating laser beam, isobtained modifying the integration range in Eq. (8)to [ x, + ∞ ). (cid:42) b y ( r ) and (cid:40) b y ( r ) have similar definitions,swapping the role of x and y . We assume the equilib-rium in the transverse direction to be reached quickly,so the normalized density is written ρ ( x, y, z, t ) = ρ ( x, y, t )(2 πL z ) − / e − z L z , where L z is the transversesize of the cloud, assumed to be small and constant. In-serting into Eqs. (7) and (8), and averaging over thetransverse direction with weight (2 πL z ) − / e − z L z , oneobtains an expression for the effective force in 2 D : F ( r ) = (cid:18) − C (cid:82) sgn( x − x (cid:48) ) ρ ( x (cid:48) , y ) dx (cid:48) − C (cid:82) sgn( y − y (cid:48) ) ρ ( x, y (cid:48) ) dy (cid:48) (cid:19) , (9)with C = (cid:126) k Γ2 s N √ πL z σ (1 + 4¯ δ ) . (10)The friction force due to Doppler cooling associatedwith the LRI beams is given by F d = − mη v where thefriction coefficient η is given as in Ref. [21] in the lowsaturation limit by η = − s (cid:126) k m ¯ δ (1 + 4¯ δ ) . (11)The friction is typically strong enough to warrant anoverdamped description of the atomic cloud (see Ref.[21]for a detailed discussion).We introduce the two parameters τ = η/ω and L = (cid:112) C/ ( mω ), with ω the in-plane harmonic trap frequencyin the xy plane, and the dimensionless variables: s = t/τ , r (cid:48) = r /L , ρ (cid:48) = ρ L , and F (cid:48) D = F D /mω .Then, taking into account the 2D approximation of theforce due to the shadow effect Eq.(9), the trapping forceand the temperature, the overdamped equation of motioncan be expressed as a continuity equation (see [22] for adetailed derivation): ∂ρ (cid:48) ∂s = ∇ (cid:48) · [ − ρ (cid:48) ( F (cid:48) D ( r (cid:48) ) − r (cid:48) ) + T ∇ (cid:48) ρ (cid:48) ] , (12)where T = k B Θ C is the dimensionless temperature, andwe get ∇ (cid:48) · F (cid:48) D ( r (cid:48) ) = − ρ (cid:48) . (13)We note that the system of Eqs. (12)-(13) has a form sim-ilar to the Smoluchowski-Poisson system of Eqs. (4)-(5).Indeed, the divergence of the interaction force is the samein both cases, but the long-range force is now anisotropicand does not derive from a potential. Since Eq.(9) hasthe form of a 1D gravitational interaction along each axis,we shall refer to Eqs.(12)-(13) as the ”1D+1D” gravita-tional model. The additional harmonic force ensures thestability of the high-temperature phase, as already men-tioned in the previous section.It is shown in Ref. [9, 23] that the system of Eqs.(12)-(13) is stable at high temperatures: the diffusionwins over the attraction and equilibrates the externalharmonic confinement. It is also suggested in [9] thatthe system undergoes a collapse transition in a finite timefor T < T c , where T c is the critical temperature of thetransition. Numerical simulations allow us to estimatethe transition temperature and give T c ≈ . − .
15, tobe compared with the 2D gravitational model for which T c = 1 / (2 π ) (cid:39) . (a)(b) Collapsed phaseExtended phase (units of )
FIG. 2. (a) Phase diagram of the 2D collapse transition.Above the curves, the model predicts a collapsed phasewhereas the extended phase lies below. We use Eq. (10)with a critical dimensionless temperature of T c = 0 .
14 anda temperature θ = 1 µ K. (b) The expected compression fac-tor CF in the collapsed phase (see text for more details). Thefull, dashed and dotted-dashed curves correspond to an atomsnumber of N = (3 , , × , respectively. Parameters ofEq.(14) are chosen from the experimental setup (see text).The vertical black line indicates ¯ δ = −
3, the detuning usedin the experiment. red curves correspond to critical lines for atoms numberof N = (3 , , × respectively. Parameters are cho-sen according to the experiment: the gas temperature is θ = 1 µ K and the trapping frequencies are ω = 20 Hz,and ω z = 300 Hz in the horizontal plane, and along thevertical axis, respectively. The critical lines are obtainedsetting a dimensionless critical temperature T c = 0 . δ = − s (cid:39) .
2, the exact valuedepending on the atoms number. Those parameters areeasily achieved in the experiment.We now discuss what should be the experimental sig-nature of this collapsed regime. The expression Eq. (13)for the attractive long-range interaction force relies onthe linearization of the laser absorption. In particular, itis not valid anymore as soon as the optical depth reachesvalues of order one, which will happen as the cloud con-tracts. In this case, the shadow effect becomes weakernear the center of the cloud, and not long-range any-more; we then expect a finite compression of the cloudwith an optical depth saturated to a value of order one.Setting the peak optical depth to one provides an orderof magnitude of the cloud’s size in the collapsed phase L = N πL z σ (1 + 4¯ δ ) . (14)For the sake of simplicity, we have considered here anisotropic cloud in the xy plane with Gaussian profiles.We define a compression factor asCF = L th /L, (15)where L th = ω − (cid:112) k B θ/m is the cloud’s size in a har-monic trap of frequency ω at thermal equilibrium, with-out long-range force. Expected compression factors inthe collapsed phase correspond to the curves in Fig.2b. Importantly, the compression factor increases asthe atoms number decrease. Thus, in the experiment,where atoms losses are present (see Sec. III C), the col-lapsed phase is expected to be characterized by an in-creasing compression factor together with a saturated op-tical depth, as time increases (and atoms are lost). Whentoo many atoms are lost, the system should finally leavethe collapsed phase and its size increases.We note in Figure 2 (b), that CF can be smaller thanone (see region where | ¯ δ | is small). It means that thecloud in the harmonic trap has an optical depth largerthan one without long-range force. In this situation, theLRI lasers are not expected to play a significant role.Therefore, in this region, Eq.(14) is no longer valid andCF must be replaced by one. III. EXPERIMENTA. Cloud preparation
The atomic system consists in a laser cooled atomiccloud of Sr [24–27]. The detailed two-stages coolingin a magneto-optical trap (MOT) is presented in Ref.[28]. The last stage of the cooling scheme as well asthe 2D artificial gravity are obtained with red detunedlasers addressing the S → P intercombination line ofnatural linewidth Γ = 2 π × . λ = 689 nm.After the final cooling stage, atoms are transferred intoa single beam horizontal optical dipole trap (ODT) at925 nm linearly polarized along ˆ z . The quantization axisis taken along the ODT beam polarization, i.e. alongthe vertical axis. The wavelength and polarization of theODT beam are chosen such that the transitions m = 0 → m (cid:48) = ± xy plane (see Fig. 1) and thus,address the transitions m = 0 → m (cid:48) = ± w Y = 138 µ m and w z = 14 µ malong the axes ˆ Y and ˆ z respectively, whereas the Rayleighlength -along ˆ X - is 360 µ m (see Fig. 1). The ODT poweris 0 .
95 W, leading to a trap depth of about 25 µ K andtrapping frequencies of 11 . X , ˆ Y , and ˆ z axes respectively. We load typically3 × atoms at a temperature of 1 µ K, leading to cloudsizes, at equilibrium and without the LRI beams, around L X = 140 µ m, L Y = 50 µ m and L z = 5 µ m. The tem-perature remains almost constant in the xy plane. In ad-dition, a dimple trap beam at 852 nm propagating alongˆ z allows for further trapping in the horizontal plane. Thedimple beam has a waist of 80 µ m and power 180 mWat the level of the cloud. The role of the dimple trapconsists in reducing the size of the atomic cloud, helpingto reach the stationary regime of artificial gravity exper-iment in a shorter time. The dimple beam is switched offwhen the LRI beams are turned on.The intensity of each LRI beam is balanced indepen-dently using half wave plates and polarizing beamsplit-ters. Intensity balance is realized when the cloud’s centerstays at a fixed position all along the experiment.The experiment is done by varying s in the range of0.2 to 2, and for each s , the duration of the LRI beams(before imaging) is varied up to 100 ms. B. Imaging scheme
The analysis of the atomic cloud is performed thanksto a fluorescence imaging system having its optical axisalong the vertical axis (see Fig. 3). The probe laser istuned on resonance with the dipole allowed transitionat 461 nm, for optimal signal-to-noise ratio. The probebeam makes a 30 ◦ angle with the ˆ x axis. Due to the lowoptical depth of the cloud along the ˆ z direction, the in-tegrated fluorescence signal is proportional to the atomsnumber. The coefficient of proportionality is extractedthanks to a preliminary joint absorption and fluorescenceimaging measurement. The atomic cloud is fully charac-terized using three fluorescence images. A first one, athigh saturation intensity, is taken after turning off theLRI beams. This image allows us to extract the cloudsize in the horizontal plane. Since the saturation is high,the absorption of the probe is weak, which gives a pre-cise ( i.e. a relative statistical error below 10%) estima-tion of the atoms number (see a sample image in rightupper panel in Fig. 3). A second image, at low satura-tion, is also taken after extinction of the LRI beams. Inthis case, absorption of the probe is clearly visible (seea sample image in right lower panel in Fig. 3), allowing s img = img = zEMCCDstaged telescopeirisProbe science chamber FIG. 3. Schematic of the imaging system and fluorescenceimaging . The blue dotted arrow shows the direction of prop-agation of the resonant 461 nm probe. Right panels: The falsecolor fluorescence images are obtained for saturating (upper)and non-saturating (lower) probes. The counts of the non-saturated fluorescence image is multiplied by two for read-ability. s img corresponds to the saturation parameter of theprobe. for optical depth measurements. With those two images,we perform a full characterization of the sizes and opticaldepths of the atomic cloud.Importantly, the measured optical depths correspondto the 461 nm transition and thus, need to be transposedto the 689 nm transition of interest. Since, the lattertransition is narrower, Doppler broadening shall be con-sidered [28]. To do so, we measured the temperature ofthe cloud using a third image at high saturation takenat long time (typically 300 ms) after turning off the LRIbeams. In this case, the cloud has reached thermaliza-tion in the ODT, so the horizontal temperatures can beextracted from the cloud sizes L th i ( i = X, Y ), and thetrapping frequencies. Moreover, comparing the cloud sizebefore and after thermalization time gives access toCF i ( t exp ) = L th i L i ( t exp ) . (16)the compression factor experienced by the cloud due tothe 2D gravity effective interaction. Here, t exp corre-sponds to the duration of the 2D gravity experiment. C. Atoms losses
In Fig. 4, we plot the lifetime of the cold atomic cloudin the ODT as a function of the saturation parameter s of the long-range force laser beams. Without the LRIbeams, the lifetime is above 20 s. The strong reduc-tion of the cloud lifetime, when the LRI beams are on,is due to atoms spilling out of the trap along the un-cooled vertical direction. When compression occurs in s t / e ( m s ) FIG. 4. (left) Atom 1 /e -lifetime in the presence of 2D gravitybeams for ¯ δ = − N (cid:39) × from data exponentialfit (red diamond) and analytical expression of Eq. (18) (solidline). the xy plane due to long-range 1D+1D forces, the tem-perature is expected to remain approximately constantin-plane because of the optical molasses, but to increasealong the vertical direction ˆ z . As atoms gain mechani-cal energy, they overcome inevitably the trap depth U at some point and are removed from the system. Weprovide now a simple model quantifying this effect. Weconsider the escape process as a single particle problemand discard the spatial distribution of atoms in the trap.At the beginning of the gravitational experiment, atomsare thermalized, and their temperature is much less thanthe trap temperature U /k B . Hence, we take the initialatom energy to be zero and consider its increase due tospontaneous emission. We assume each scattering eventincreases the kinetic energy of an atom in the verticaldirection by E r , of the order of the photon recoil energy.Since n = U E r (cid:39)
110 is large, each atom will undergomany scattering events before leaving the cloud, and thisapproximation should be reasonable. The effective scat-tering rate is given by (see for instance [31]) ξ = 125 Γ s δ . (17)The prefactor comes from the radiation pattern and num-ber of LRI beams. Since the photon scattering rate ξ is constant, the number N S ( t ) of scattering events fol-lows a Poissonian distribution with parameter ξt , andthe energy of an atom is E ( t ) = E r N S ( t ). At time t ,the fraction of atoms remaining in the trap is then givenby P ( N S ( t ) < n ) = F ( ξt, n ), where F ( λ, n ) is the cu-mulative distribution function of a Poisson variable withparameter λ . The characteristic 1 /e -lifetime correspond-ing to N ( t /e ) = N ( t = 0) /e can be obtained using theStirling’s formula for the incomplete Gamma function;one gets t /e ≈ nξ . (18)Fig. 4 gives a very good agreement between the ex-periment and the model of Eq. (18) when evaluating the s C F X C F Y b X - Y - b t (ms) t (ms) FIG. 5. Optical depths at detuning ¯ δ = − X and ˆ Y fora temperature θ ≈ µ K, and an initial atoms number N ≈ × . characteristic 1 /e -lifetime of atoms within the trap. Thisquantitative evidence suggests that indeed single atomheating and spilling along the vertical direction is at theorigin of the atom losses. D. Experimental results
The temporal evolution of the optical depth and com-pression factor along ˆ X and ˆ Y (the ODT proper axes) forvarious saturation parameters are given in Fig. 5. Ac-cording to the 2D model (see Sec. II C and Fig. 2), thesystem should be in the collapsed phase, at least for thelarge values of the saturation parameter. However, wedid not observe the expected signatures of the collapsedphase which are an increasing of the compression factorand a saturation of the optical depth to a value aroundone. Indeed, after a short time of 5 −
10 ms we observe acompression of the cloud of about 60% in the ˆ Y directionand a more moderate compression of about 30% in theˆ X direction. As expected, those compression factors arehigher for larger saturation parameter, but the compres-sion factor rapidly falls to value close to one. The opticaldepth (left column) is initially rather large, which mightexplain the initial moderate compression. However, weobserve a monotonous decrease of the optical depth with-out any sign of saturation. We observe that the decreaseof the optical depth is more pronounced at large satura-tion in agreement with a larger atomic loss rate (See Fig.4). The compression factors larger than one at t = 0originate from the vertical dimple trap beam as shownby the green arrow on Fig. 1. This beam is turned off at t = 0. IV. THREE-DIMENSIONAL MODEL
In the previous section, the experimental results showa moderate compression of the atomic cloud, but no sig-nature of a collapsed phase as predicted by the 2D modelof Sec. II A. In order to be closer to the experimental re-ality, we generalize this model in 3D, including the finitethickness of the cloud in the vertical direction, and theCoulomb-like repulsion induced by multiple scattering.
A. Description of the model
For a 3D model, the particle dynamics is now drivenby three forces depending on the particle position, F tot = F + F T + F M and by F d , the friction force associatedwith Doppler cooling. F is the attractive force due to the LRI beams, F T the harmonic trapping force, and F M is the repulsiveforce coming from multiple scattering, which cannot bediscarded in a three dimensional geometry. The expres-sion of the attractive force is: F ( r ) = − C (cid:48) (cid:82) dx (cid:48) sgn( x − x (cid:48) ) ρ ( x (cid:48) , y, z ) (cid:82) dy (cid:48) sgn( y − y (cid:48) ) ρ ( x, y (cid:48) , z )0 , (19)where C (cid:48) = (cid:126) k Γ2 s N σ (1 + 4¯ δ ) . (20)Here, the LRI beams lead to the same force than derivedin 2D.To mimick the experiment, we consider an anisotropictrap, with its principal axes along the unit vectors ˆ X , ˆ Y ,ˆ z ( ˆ X, ˆ Y are not aligned with the LRI beams, see Fig.1).The associated force at a position R = X ˆ X + Y ˆ Y + z ˆ z is: F T ( R ) = − m (cid:104) ω X X ˆ X + ω Y Y ˆ Y + ω z z ˆ z (cid:105) . (21)The friction force F d has the same expression as in the 2Dmodel. Finally, the repulsive force coming from multiplescattering is given by [3] F M ( r ) = D (cid:90) d r (cid:48) r − r (cid:48) | r − r (cid:48) | ρ ( x (cid:48) , y (cid:48) , z (cid:48) ) , (22)where D = (cid:126) k Γ2 s N σ σ R π (1 + 4¯ δ ) . (23)and σ R is the re-absorption cross-section of scatteredphotons. We assumes an isotropic fluorescence patternand multiple scattering limited to a single re-absorptionevent. The latter is well justified in the low optical depthregime. The former overestimates the multiple scatter-ing contribution with respect to the experiment wherethe LRI beam polarization are in the horizontal plane,leading to a radiation pattern more pronounced (by afactor of two) along the vertical direction.To make the comparison between the strength of theattractive and repulsive forces easier, we write the equa-tions in a slightly different way than in Sec.II A. By in-troducing the length scale LL = 2 (cid:126) k L Γ σ R mω , (24)and the timescale τ τ = (cid:16) ηω (cid:17) , (25)where ω is a characteristic trap frequency in the xy plane(the actual frequency is not the same along the X and Y axes). Then the Smoluchowski equation can be expressedin a dimensionless form ∂ρ∂t = ∇ · [ − ρ F tot + T ∇ ρ ] , (26)where the dimensionless temperature T is now given by T = k B θmω L = k B θ (2 √ mω (cid:126) k L Γ σ R ) / , (27)and the dimensionless forces can be written as F ( r ) = − γc (cid:18) (cid:82) dx (cid:48) sgn( x − x (cid:48) ) ρ ( x (cid:48) , y, z ) (cid:82) dy (cid:48) sgn( y − y (cid:48) ) ρ ( x, y (cid:48) , z ) (cid:19) , F M ( r ) = c π (cid:90) d r (cid:48) r − r (cid:48) | r − r (cid:48) | ρ ( x (cid:48) , y (cid:48) , z (cid:48) ) , (28) F T ( R ) = − (cid:20)(cid:16) ω X ω (cid:17) X ˆ X + (cid:16) ω Y ω (cid:17) Y ˆ Y + (cid:16) ω z ω (cid:17) z ˆ z (cid:21) . The parameters c and γ are given by c = N s (1 + 4¯ δ ) and γ = σ σ R , (29)Note that, in the low saturation regime, the scatteringof re-emission is elastic (no change of photon frequency)meaning that γ = 1, reaching its maximal value. If sat-uration of the transition occurs, the scattering becomesinelastic and part of the fluorescence spectrum is broughtat the transition resonance [32], increasing σ R . Follow-ing [33], we estimate γ ≈ .
98 for a saturation parameter s = 1 per beam. This is the value we have used in thesimulations. B. Numerical simulations
We have performed two types of simulations of Eqs.(26),(28): i) with a fixed number of atoms; ii) includingatoms losses in an effective manner: interaction forces t (ms) s = s = s = C F Y C F X t (ms) s = s = s = FIG. 6. Time evolution of the compression factors along thetwo directions ˆ X (up) and ˆ Y (bottom), obtained by directsimulation of the two 3D models: The dashed curves corre-spond to the model with a constant number of atoms and thefull curves to the model including effective atoms losses. Theangle between the axes ˆ x and ˆ X is set to 30 ◦ (to be comparedto 23 ◦ in the experiment). The corresponding dimensionlesstemperature of the 2D model for s = 1 is ∼ . i.e. in thepredicted collapsed phase according to Sec.II A. then have an exponentially decreasing strength c ( t ) = c e − t/t /e ; we have taken t /e as given by the curve inEq. (18).A proper numerical integration of the Smoluchowski-Poisson equation, given by Eq. (26), is a challenging task.The repulsive force is computed via a standard Poissonsolver for the Coulomb-like potential. For the attrac-tive force, we note that, although it does not globallyderive from a potential, the x and y components of theforce taken separately do. We then use the finite-volumemethod presented in [34], which is well suited to obtainsolutions at long times for such potential forces, and wecouple it with a splitting procedure: we first computea time step with only the x component of the attrac-tive force, then a time step with only its y -component,and repeat. A further numerical difficulty is related tothe spatial scale difference between the xy plane and thetransverse z direction.Because the compression observed in experiment is atransient phenomenon, we focus our study of the 3 D m a x ( C F ) FIG. 7. Maximum transient compression factors alongthe two trap directions (red squares) ˆ X and (blue stars)ˆ Y as a function of the normalized trap ratio for s = 1. ω z / √ ω X ω Y = 15 corresponds to the experimental value, in-dicated by the vertical dashed line. Simulation results corre-spond to the model where the number of atoms is constant.The shift between the compression factors between the X -direction and the Y -direction are due to the trap anisotropyin the horizontal plane. models on the time evolution in order to compare moreefficiently the numerical results to our experiment. Wefirst consider the time evolution of the compression fac-tors for the two 3D models (see Fig. 6). The values of thesimulation parameters are chosen in agreement with theexperimental ones. The dashed curves correspond to aconstant atoms number, and the plain curves include anexponential decay of the atoms number, with t /e chosenas in Eq. (18), for three different values of the saturationparameter s = 0 . , ,
2. The initial number of atomsis N = 3 × , the detuning is ¯ δ = −
3, and temper-ature is θ = 1 µ K. The planar anisotropy of the trap isset by the ratio ω z / √ ω X ω Y = 15, and the ratio of thetrap frequencies in the xy plane is ω Y /ω X = 2 .
6. Sincethe initial density distribution ρ ( t = 0) is chosen as anisotropic Gaussian in the xy plane, with thermal widthcorresponding to ω X , the initial value of the compressionfactor along the X axis is CF X ( t = 0) = 2 . t ∼ −
20 ms, and then decreases. However, simulations predict larger compression factors than thoseobserved. The discrepancy between the model and theexperiment may come from the assumptions made in the3 D model: the long range of the shadow force is associ-ated with the linearization of the Lambert law, leading toa stronger compression force. At longer time ( >
20 ms),atoms losses drive the system towards a trivial stationarystate, without compression.When the number of atoms is kept constant, (dashedcurves in Fig. 6), we observe the same behavior at shorttime, i.e. when the atoms losses are not significant. Atlong time ( (cid:38)
20 ms), the system settles in this case ina stationary compressed state. Comparing with the 2 D model, we conclude that the 3D effects play a significantrole to explain the relatively small observed values of thecompression.We now investigate the role of the trap aspect ratio ω z / √ ω X ω Y ; to do so, we keep constant the initial op-tical depth in the plane z = 0, which amounts to keepconstant the rescaled temperature T in the associated 2 D model, well inside the ”collapse region”. Fig. 7 displaysthe compression factor versus the trap aspect ratio. Weexpect that the behavior of the system becomes closerto the prediction of the 2 D model when this aspect ra-tio tends to infinity. However, the 3D model predictsthat the size of the system saturates to a finite value,in contrast to the infinite compression predicted by thecollapsed phase of the 2 D model.To illustrate the time evolution, Fig. 8 shows severalsnapshots of the atomic spatial distribution in the z = 0plane at three different times: t = 0, t = t m = 6 ms cor-responding to the maximum compression along ˆ X and atthe final time t = 100 ms of the simulation. We first focuson the upper row which corresponds to the case of con-stant atom number. We observe a fast compression of thecloud as also indicated by the compression factor in Fig.7; the cloud at t = t m has a long-range star-shape simi-lar to the one observed in the 2 D simulations in [9], cyandash-dotted contour line helps to visualize this shape. At t = 100ms, the system has essentially reached a station-ary state, but it displays unexpected spatial patterns:the cloud has split into two parts along one bisector ofthe LRI lasers, where the attractive long-range force isthe weakest. This rich dynamic results from a competi-tion between the geometry-driven attractive and repul-sive long-range interaction. A detailed study of this phe-nomenon is beyond the scope of paper. We focus now onthe lower row of Fig. 8, corresponding to the case withatoms losses. As already discussed, the effect of interac-tions becomes negligible at large times, a decompressionoccurs and the cloud’s shape corresponds to the expectedshape in the harmonic trap at thermal equilibrium. V. CONCLUSION
We have studied the interaction of a quasi-two-dimensional ultra-cold atom cloud with two orthogo- nal quasi-resonant counter-propagating pairs of lasers.For low optical depth, each pair of laser mimics one-dimensional artificial gravity-like long-range force. Wehave reached experimentally the regime where a two-0 x ( a . u . ) x ( a . u . ) x ( a . u . ) x ( a . u . ) x ( a . u . ) FIG. 8. Simulated spatial distributions of atoms in the plane z = 0 at t = 0 (left), at t = 6 ms (middle) correspondingto the maximum of the compression factor along ˆ X , and at t = 100 ms (right). s = 1. Upper panels (top green dottedarrow) correspond to the model with a constant number of atoms and lower panels (bottom red dotted arrow) to the modelwith an exponential decrease of the atom number. We also draw the principal axes of the trap (white dashed) and the firstbisector angle between LRI lasers (red continuous). Contour lines, for intermediate time, highlight the star-shape of the cloudas predicted by the 2D model (see [9]). a.u. stands for arbitrary units. dimensional analysis predicts a collapse of the cloud, butwe have observed only a moderate compression. To un-derstand this discrepancy, we have introduced a three-dimensional model which provides a more realistic de-scription of the cold atom cloud, and in particular in-cludes the repulsive long-range force coming from pho-tons reabsorption: our results show that although repul-sive long-range force is partly suppressed by the pancake-shape geometry as expected, it is non negligible whenthe compression occurs. If we include atoms losses alongthe uncooled vertical dimension, the model is in qual-itative agreement with the experiment. We conjecturesome of the remaining discrepancies may be due to thelow optical depth approximation which is implemented in the theoretical approaches. 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