Production of Sodium Bose--Einstein condensates in an optical dimple trap
David Jacob, Emmanuel Mimoun, Luigi De Sarlo, Martin Weitz, J. Dalibard, F. Gerbier
aa r X i v : . [ phy s i c s . a t o m - ph ] M a y Production of Sodium Bose–Einstein condensates inan optical dimple trap
D Jacob, E Mimoun, L De Sarlo, M Weitz † , J Dalibard and FGerbier Laboratoire Kastler Brossel, ENS, UPMC-Paris 6, CNRS; 24 rue Lhomond, 75005Paris, France † Institut für Angewandte Physik, Universität Bonn, Wegelerstr. 8, 53115 Bonn,Germany
Submitted to:
New J. Phys.
PACS numbers: 67.85.Hj, 37.10.De, 37.10.Jk
Abstract.
We report on the realization of a sodium Bose–Einstein condensate (BEC) in acombined red-detuned optical dipole trap, formed by two beams crossing in a horizontalplane and a third, tightly focused dimple trap propagating vertically. We produce aBEC in three main steps: loading of the crossed dipole trap from laser-cooled atoms, anintermediate evaporative cooling stage which results in efficient loading of the auxiliarydimple trap, and a final evaporative cooling stage in the dimple trap. Our protocolis implemented in a compact setup and allows us to reach quantum degeneracy evenwith relatively modest initial atom numbers and available laser power.
1. Introduction
The preparation of degenerate atomic quantum gases is interesting from both afundamental and an applied point of view. On the one hand, the unprecedented levelof control on these systems allows one to study quantum many-body phenomena in theabsence of perturbing effects unavoidable in solid-state systems [1]. On the other hand,degenerate gases are a promising starting point to reliably produce highly entangledstates, which could pave the way for a new generation of atom-based quantum sensors(see [2, 3] and references therein).In view of the sensitivity of these strongly correlated states to the perturbationscaused by magnetic fields fluctuations, experimental schemes in which evaporativecooling is performed without the use of external magnetic fields are particularlyinteresting. These so-called “all-optical evaporation” schemes rely on far off-resonantoptical dipole traps. They have been developed by several groups to produce Bose–Einstein condensates (BEC) of various atomic species, in particular alkali atoms (Rb roduction of Sodium Bose–Einstein condensates in an optical dimple trap N in the crossing region and the temperature T at the end of the freeevaporation to get a high phase-space density D and a large collision rate γ coll . We recallthat D = N ( ~ ω/k B T ) and γ coll ∝ N/ω T for a Boltzmann gas in a harmonic trap,with ~ the Planck constant, k B the Boltzmann constant, and ω the average trappingfrequency. For a given beam size, we find that the optimal trap depth is different forloading and free evaporation, and propose that trap-induced light-shifts on the coolingtransition are the physical mechanism behind this observation.The second issue is related to the efficiency of evaporative cooling. In theserespect, optical traps differ in several aspects from magnetic traps. In magnetic traps,evaporative cooling takes place in the so-called runaway regime, where the elasticcollision rate γ coll and evaporation efficiency stay constant or even increase with time[13]. In optical traps, this regime is not easily reachable because the trap depth and trapconfinement both increase with the trapping laser power. In practice, decreasing thetrap depth to force evaporation results in a looser confinement, so that the collision ratecan decrease even if the phase-space density increases. Solutions involving modificationof the trapping potential have been demonstrated to resolve this issue. For example adynamical change of the beam size using a zoom lens allows one to maintain constantconfinement while reducing the trap depth [5], thus preserving a high collision rateduring evaporation. Runaway evaporation in an optical trap can also be obtained,by using an additional expelling potential independent of the trapping laser (gravityor “pulling” laser) in order to decouple trap confinement and potential depth [14, 15].A third solution, based on the addition of a tighter “dimple” potential [8, 16], hasbeen realized and characterized theoretically [17, 18, 19]. This solution, which is theone investigated in this paper, leads to a two-step evaporation sequence: After theloading of a larger trap, atoms are first transferred by cooling into the “dimple” trap, roduction of Sodium Bose–Einstein condensates in an optical dimple trap Na. Starting with × trapped atoms,a Bose–Einstein condensate of ∼ Na atoms is produced after ∼ s evaporationtime.The paper is organized as follows. In section 2, we give an overview of ourexperimental setup. In section 3, we investigate the loading of a dipole trap from aMOT of sodium atoms and study how the compression of the trap after the atom captureimproves the initial conditions for evaporative cooling. We then present in section 4 howevaporative cooling works in presence of the auxiliary dimple trap, detailing its fillingdynamics, and the last evaporative cooling stage to reach Bose–Einstein condensation.
2. Experimental Setup
Our experiment starts with a sodium magneto-optical trap (MOT) capturingapproximately atoms in s from a vapor whose pressure is modulated usinglight-induced atomic desorption [20]. After the MOT is formed, a far off-resonantdipole trap is switched on (see subsection 2.2). The detunings and powers of boththe cooling (tuned to S / , F = 2 → P / , F ′ = 3 transition) and repumping (tuned to S / , F = 1 → P / , F ′ = 2 transition) lasers are modified in order to optimize the traploading. During a first “dark MOT” phase [21], we lower the power of the repumpinglaser in about ms, from I rep = 300 µ W.cm − to I rep = 10 µ W.cm − per beamwhile keeping the magnetic gradient on. This reduces the loss rate due to light-inducedcollisions by limiting the population of the excited states [11]. We keep the cooling laserintensity at the same value as for MOT loading, I cool = 0 . mW.cm − per beam, whichcorresponds to one sixth of the saturation intensity ( I sat = 6 . mW.cm − ). During this“dark MOT” phase, both the spatial density in the dipole trap and the temperatureincrease. We then apply a ms-long “cold MOT” phase, where the cooling beamdetuning is shifted from δ cool ≈ − Γ to δ cool ≈ − . ( Γ / π ≈ MHz is the naturallinewidth). The temperature of the atoms after this cooling sequence is around µ K. The far off-resonant dipole trap results from the combination of three beams, twoforming a crossed dipole trap (CDT) in the horizontal x − y plane and a tightly focusedone propagating vertically along the z axis (see figure 1(a)), which we refer to as “dimpletrap” (dT). The CDT is derived from a W fiber laser (IPG Photonics) at nm.This trap is formed by folding the beam onto itself at an angle θ ≃ ◦ in the horizontalplane. At the crossing point, both arms have a waist w CDT ≈ µ m. We control roduction of Sodium Bose–Einstein condensates in an optical dimple trap ω x / π (kHz) ω y / π (kHz) ω z / π (kHz) V /k B (mK) CDT . . . . dT . . .
021 0 . Table 1.
Trapping frequencies and trap depths at P CDT ≈ W and P dT ≈ mWfor the crossed dipole trap (CDT) and the dimple trap (dT), respectively. the laser power using a motorized rotating waveplate (OWIS GmbH) followed by aGlan-Taylor polarizer (bandwidth ∼ Hz), and a control input on the current in thelaser pump diodes (bandwidth ∼ kHz). The waveplate is used for coarse reductionof laser power by changing the amount of light transmitted by the polarizer, whereasthe current control is used at the end of the evaporation ramp (low laser powers) andfor fast servo-control of the intensity to reduce fluctuations. Combining both servoloops, we can control the laser power from its maximal value ( P CDT ≈ W) down to ≈ mW. We can switch off the trapping potential to an extinction level greater than
90 % in less than µ s using the laser current input. We use motorized mirrors (Agilis,Newport Corporation) for alignment. Special care is taken to ensure the orthogonalityof the polarization of both arms, realized by the insertion of a λ/ waveplate that ispositioned with a precision . . ◦ . A misalignement of only ◦ results in a measurableheating of the sample [20].The auxiliary dimple trap is produced using a mW laser (Mephisto-S, InnoLightGmbH) at nm. As sketched on figure 1(a), the beam propagates vertically, andcrosses the CDT with a waist w dT ≈ µ m. The laser beam is transmitted througha single-mode optical fiber, and focused to a waist size w dT using a custom-mademicroscope objective (CVI Melles Griot, NA & . ). An acousto-optic modulator placedbefore the fiber allows us to control the intensity and to quickly switch off the dT-beam.To fix the notations that will be used in the following, we give here the expressionsof the dipole trap potentials. The expression of the CDT potential is given by V CDT ( x, y, z ) = − V " e − x + z ) /w ( y ) ( w ( y ) /w CDT ) + e − u + z ) /w ( v ) ( w ( v ) /w CDT ) , (1)with w ( y ) = w CDT p y /y R and with y R the Rayleigh length y R = πw /λ ≈ . mm. We have also introduced the rotated coordinates: ( u, v ) = ( x cos( θ ) + y sin( θ ) , − x sin( θ ) + y cos( θ )) . The expression of the dT potential is given by V dT ( x, y, z ) = − V e − x + y ) /w , (2)neglecting the confinement of the dT along the z -axis, always negligible compared tothe vertical confinement of the CDT. Typical trapping frequencies and potential depthsare given in table 1.In figure 1(c), we give schematically the temporal evolution of the powers of the twolasers during the experimental sequence. This time evolution is optimized for loadingand evaporation, as explained in sections 3 and 4. roduction of Sodium Bose–Einstein condensates in an optical dimple trap We monitor the time evolution of the trapped cloud using both fluorescence andabsorption imaging [22]. Fluorescence light is captured by the same high numericalaperture microscope used to focus the dT. The photons are collected on a low-noisecharge-coupled device camera (PIXIS, Princeton Instruments). In figure 1(b), we showa typical fluorescence image. We typically observe the atoms after a time-of-flight t ToF = 0 . ms during a short pulse ( t mol = 50 µ s) performed with the six coolingand repumping beams.Absorption images are recorded with a vertically propagating resonant probe beam.It is better suited for the analysis of the central denser part of the trapped cloud butnot very precise for the arms of the CDT. Indeed, the regions corresponding to the CDTarms display low optical densities ( < . ) only slightly above the noise level ( ∼ . ,limited by residual fringes on the background). Atom counting in the arms of the CDTis thus more accurate using fluorescence images.
3. Loading and free evaporation in the Crossed Dipole Trap
We can distinguish two stages in the dynamics of the trap loading. At first, duringthe MOT/CDT overlap period, atoms are captured mainly in the arms of the CDTwithout a notable enhancement of the density in the crossing region. In the secondphase that follows the extinction of the MOT beams, which we call “free evaporation”,the hottest atoms leave the arms and the remaining ones fill the crossing region throughthermalization. The quantity of interest is the number of atoms in the central region N C , which corresponds approximately to the number of atoms with an energy comprisedbetween − V and − V / (as defined in equation (1)). We show in figure 2 thepotential V CDT in the z = 0 plane, truncated at three different energy levels. One cansee that atoms having energies lower than V / explore only the central region, asexpected. This dense part is the relevant component that matters for further evaporativecooling. Although both trapping lasers (CDT and dT) are turned on simultaneously, theCDT is much deeper than the dimple trap, the latter playing a negligible role during thisinitial stage. In this section, we discuss auxiliary experiments where the dT is absent.In order to understand the loading dynamics during the first stage, we give a briefoverview on the relevant mechanisms (see [11] for a detailed analysis). The loading rateof atoms in the CDT is proportional to the probability of an atom to be trapped by thedipole potential and to the atomic flux in the CDT/MOT overlap region. The first termcorresponds to the damping of the velocity of an atom when it crosses one arm of theCDT, leading to a reduction of its total energy below the CDT potential depth. Thesecond term is proportional to the spatial density and the average velocity of the atoms inthe MOT, and thus depends on the temperature of the atoms. The relevant parametersfor optimizing the loading rate, namely the atomic density and the temperature, can be roduction of Sodium Bose–Einstein condensates in an optical dimple trap w CDT (from µ m to µ m). A larger beam size helps to trap more atoms during the capturestage due to higher overlap volume. However, at a given available power, larger beamsimply a weakening of the trap stiffness, which in return penalizes the thermalization aftercapture. The data presented in this paper are taken with a beam waist w CDT ≈ µ m.We obtain very similar results for w CDT ≈ µ m, but with different optimal powers ateach stage. In the next subsection, we will concentrate on the optimization of the laserpower to find the optimal trap depth for filling the center region. In order to characterize the filling dynamics of the crossing region, we define the fillingfactor α = N C /N as the fraction of atoms in this region relatively to the total numberof atoms in the dipole trap. Images as that in figure 1(b) are processed with a multi-component fitting routine that extracts the temperature, the density, the total atomnumber N and N C . Details about the fitting procedure are presented in Appendix A.The results of the optimization of the CDT power are presented in figure 3, where weplot the evolution of atom number N C and filling factor α with time. We fit the function α ( t ) = a (1 − e − t/τ ) + b to our data.We look first at a situation in which “free evaporation” occurs at constant CDTpower, keeping the same power during the free evaporation phase as during the capturestage. We report in figure 3 the evolution of N C and α with time for three differentpowers ( P CDT = 7 . , . and W). The values of the loading time τ and the asymptoticvalue α ∞ = a + b of the filling factor obtained from the fit are shown in figure 4. Wefind an optimal power P CDT = 13 . W that maximizes both the number of atoms N C and the stationary filling fraction α ∞ .In a second set of experiments, the CDT is kept at constant P CDT = 13 . W duringthe “cold MOT” phase, and ramped up in ms to another value just after switchingoff the resonant lasers. As shown in figure 4, ramping up the power to the maximumavailable power results in quicker loading of the central region ( ≃ sec) and betterfilling ratio ( α ∞ ≃ . ), the best values being apparently limited by the available laserpower. A slower, linear power ramp to P CDT = 36
W in sec (also shown in figure 3),leads to a slightly better loading ratio ( α ∞ ≃ . ), and also a slightly lower temperature,which altogether results in a higher number of atoms (about twice as many atoms in roduction of Sodium Bose–Einstein condensates in an optical dimple trap P CDT = 13 . W). Thisparticular ramp provides the best starting point we could achieve for the evaporativecooling stage.The results of the two series of experiments show the existence of an optimal power P optCDT = 13 . W for the loading of the atoms during the period in which the MOT andthe CDT are simultaneously present. We interpret this observation in the followingway. The CDT laser exerts different light shifts on the various hyperfine states in theground ( s ) and excited ( p ) manifolds. These differential light shifts can perturb thelaser cooling dynamics in the CDT region and thus degrade the capture efficiency. Forinstance, if we take the | g i = | F = 2 , m F = 2 i → | e i = | F ′ = 3 , m F = 3 i transitionand a π -polarized CDT laser, we obtain that near the trap bottom, the laser detuningchanges according to δ = ω L − ω + α I, (3)with ω L the cooling laser frequency, ω the “bare” transition frequency, I =2 P CDT /πw the intensity of the CDT laser ‡ . For sodium, we find α / π ≈ Hz.cm /W. For our optimal cooling beam detuning δ cool = ω L − ω ≈ − . (see section 2.1) we obtain that the detuning on the cooling transition vanishes when I ≈ | δ cool | /α ≈ . × W . cm − . Experimentally, the optimum P optCDT = 13 . Wcorresponds to I opt = 4 . × W . cm − , close to the value calculated above, and achange of detuning from − . to δ ≈ − . . We reached a very similar optimumin another set of experiments with w ′ CDT = 35 µ m, where we found an optimum power P ′ optCDT = 10 W corresponding to I ′ opt = 5 . × W . cm − and a comparable final detuning δ ≈ − . .One could think that tuning the cooling beam frequency further than − . on thered side of the | g i → | e i transition could help to mitigate the effect, thus increasingthe optimal power and ultimately the number of atoms captured. However, two separateeffects work against this strategy. First, this compensation is efficient only near the trapbottom and not across the whole trapping region. Second, it brings the MOT-beamscloser to resonance with neighbouring transitions that can shift in opposite ways. Forexample the | g i = | F = 2 , m F = 2 i → | e i = | F ′ = 2 , m F = 2 i transition has anintensity dependance δ = ω L − ω − α I , with ω the corresponding frequency and α / π ≈ Hz.cm /W. The latter effect is limiting for Na, which has an hyperfinestructure splitting ω − ω much smaller than heavier alkalis ( Rb and
Cs).
4. Two-stage evaporation
As pointed out in the introduction, lowering the laser intensity to reduce the trap depthfor evaporation is inevitably accompanied by a reduction of trap stiffness (near the ‡ For the calculation we use the data from the NIST atomic spectra database [23] and consider the s → p , p → d, d transitions (see also [24]). roduction of Sodium Bose–Einstein condensates in an optical dimple trap ω is proportional to √ P /w ), unlike in magnetic trapswhere the depth and confinement are independent. The resulting decrease in densityand collision rate can make the cooling due to evaporation stop at low laser power,and this is precisely what is observed in our experiment. For a harmonic trap theclassical phase-space density is given by D = N ( ~ ω/k B T ) , where ω stands for the meantrapping frequency. In a simple model where the evaporation parameter η = V CDT0 /k B T is assumed constant and where losses are neglected, the gain in phase-space densitywhen the trap depth is lowered from V CDT0 to V CDT0 /r ( r > is the reduction factor) isgiven by [25] D = D r β , β = 32 η − η + 11 η − η + 7 . (4)The starting point in our experiment (about × atoms at T ≃ µ K)corresponds to η ≈ and a phase-space density D ∼ − . According to equation(4), evaporating with a reduction factor r = 200 leads to a final phase-space density of ∼ . . In our experiment, the laser power is ramped down during evaporation accordingto V CDT ( t ) = V (1 + t/τ evap ) − α evap , (5)where the parameters τ evap = 30 ms and α evap = 1 . were optimized empirically. Evenafter optimization we have not been able to achieve a final phase-space density greaterthan ∼ − in the CDT alone † . We also observe that the collision rate after ∼ s islower than s − , and that evaporative cooling stops near this point. Such a collisionrate is too low to maintain efficient thermalization and sustain the cooling process. The decrease of evaporation efficiency presented in the previous subsection is caused bythe relation between trap depth and confinement strength inherent to optical traps. Tocircumvent this problem, one needs to break this relation. To this aim, we have addeda tight “dimple” to the initial trap [20, 8, 26, 27]. We turn on the auxiliary dT togetherwith the main CDT, but keep it at constant power P dT = 200 mW during the CDTramp § . With the “dimple” addition, the atoms feel a more and more confining potentialas they cool, in stark contrast to the situation in the CDT alone. In our experiment,we take advantage of the dissipative nature of evaporative cooling to fill such a “dimple”trap (dT). As the atoms in the CDT are evaporatively cooled, they get progressivelytrapped in the stiffer potential which results in a substantial increase in spatial density[8, 18]. Since the temperature remains the same, this translates into a huge boost inthe phase-space density. This is markedly different from an adiabatic trap compressionwhich increases spatial density but leaves the phase-space density unchanged [28]. After † The measured phase-space density is lower than the prediction of equation (4). This should beattributed to the crudeness of the model underlying this equation. In particular, three-body losses,which are important at the densities present in the CDT, are not accounted for. § Keeping the dT power high causes no modification for the CDT loading and compression. roduction of Sodium Bose–Einstein condensates in an optical dimple trap s evaporation in the CDT, the dT power is reduced to provide the final stage ofevaporative cooling (see figure 1(c)).The plots in figure 5 summarize the evaporation dynamics. We show atom number N , temperature T , dimple trap filling α d = N d /N where N d is the number of atomspresent in the dT, and phase-space density D during the ramp, and compare it tothe evaporation without dimple trap ∗ . From figure 5(c), we observe that almost allatoms accumulate rapidly (within a few ms) in the dT. At this stage, the atomsare essentially trapped by the dT in the x − y plane and by the weaker CDT in the z − direction. We therefore call this stage “evaporative filling”. At the end of it, we obtaina cold sodium gas with high phase-space density and collision rate ( γ coll ≈ × s − ),well suited to start a second evaporative cooling stage.The difference in trapping frequencies between the cases with and without dimpletrap leads to an increase of about in phase-space density at t = 1 s. We quantify theevaporation efficiency κ evap leading from the starting point ( N , D ) to ( N , D ), usingthe the definition given in [13], κ evap = − ln ( D / D )ln ( N /N ) . (6)Typical values in magnetic traps are κ evap ∼ − . In our experiment, we get muchbetter evaporation efficiencies using the dimple trap ( κ dT evap ≃ . ) than evaporation inthe CDT alone ( κ CDTevap ≃ . ) ♯ .We pursue the evaporation by reducing the dT depth, with an exponential rampfrom P dT = 200 mW to P dT = 2 mW in . sec, with a time constant τ dT = 0 . s.This results in a phase-space density increase and a crossing of the BEC threshold after ∼ sec ramping, with ≃ × atoms at T ≃ µ K. At the end of this ramp, we obtainan almost pure BEC with ≃ atoms.Finally, we note that the dimple trap is used here in a quite different way comparedto the experiment reported in [26, 27]. In these works, the authors studied an adiabaticprocess, in which the gain in phase-space density is obtained isentropically by modifyingthe trap potential shape [29]. In the present work, the entropy is reduced by evaporativecooling as the transfer between the CDT and the dT proceeds.
5. Conclusion and prospects
We have demonstrated a method to reach Bose–Einstein condensation of Na in anall-optical experimental setup. We have shown the importance of adapting the trappingpotential to the magneto-optical trap cooling dynamics for optimizing the capture in ∗ The measurement of α d is done in the following way: The dT laser is switched off . ms after theCDT laser, and we let the cloud expand during time-of-flight t ToF = 0 . ms before taking an absorptionimage. At this time, atoms released from the CDT have expanded in the x − y plane more than thosereleased from the dT, which is appropriate for counting each component. ♯ We have found experimentally that turning on the dimple trap at a later stage during the evaporationramp still results in a boost in phase-space density. However the cooling is not as efficient, so that thefinal phase-space density and the evaporation efficiency are both slightly worse. roduction of Sodium Bose–Einstein condensates in an optical dimple trap ( ω dT /ω CDT ) , where ω dT is the dT average frequency and ω CDT the CDT average frequency at low power.Experimentally this corresponds to a large gain in phase-space density, of ∼ . Aftera final evaporation stage in the dimple trap, we are able to obtain almost pure BECscontaining ∼ atoms.The efficient “evaporative filling” of the dimple trap suggests to generalize thescheme by adding a second, even smaller dimple trap to shorten the time to reachBose–Einstein condensation. Such a scheme with imbricated evaporative cooling steps(like the layers of an “atomic matryoshka”) can be taken into consideration if the aimis the production of Bose–Einstein condensates with small atom number, confined in amicroscopic potential [30, 31, 32]. Acknowledgments
We wish to thank Lingxuan Shao and Wilbert Kruithof for experimental assistance. D.J.acknowledges financial support by DGA, Contract No. 2008/450. L.D.S. acknowledgesfinancial support from the EU IEF grant No. 236240. This work was supported byIFRAF, by the European Union (MIDAS STREP project), and DARPA (OLE project).Laboratoire Kastler Brossel is a Unité Mixte de Recherche (UMR n ◦ Appendix A. Analysis of CDT images
A figure of merit for the loading in the CDT is the atom number in the crossing regionof the trap as mainly these atoms will participate to evaporative cooling. We will takethe ratio between the atoms in this region and the arms as an indicator of the loadingefficiency. We fit the atomic density profiles with a sum of three gaussians, two of themfitting the arms region and the last one fitting the center region, f ( x, y ) = X j =1 G ( A j ; x j , y j ; σ jx , σ jy ) , (A.1)with G ( A, x, y, σ x , σ y ) = A e − ( x/σ x ) − ( y/σ y ) . Here A is the amplitude, σ x and σ y the sizes of the distribution along the directions x and y . The first two components , model the arms so that σ y ≫ σ x and σ y ≫ σ x . The third componentmodels the denser crossing region. The second arm propagates with an angle θ : ( x , y ) = ( x cos( θ ) + y sin( θ ) , − x sin( θ ) + y cos( θ )) . Each arm is supposed to beradially symmetric, the size in the z direction is therefore taken equal to the radial size roduction of Sodium Bose–Einstein condensates in an optical dimple trap ( x, y ) plane. From the sizes and the calibration of the total fluorescence countson the CCD with the atom number measured from an absorption image, we infer theatom number in each component N , N and N .In order to evaluate how well equation (A.1) can fit the density profile, we performthe fit on a computed density profile of an atomic cloud at thermal equilibrium in acrossed dipole trap potential U ( r ) = V CDT ( r ) (see equation (1)), for P CDT = 13 . Wand w CDT = 42 µ m. For a classical gas, the phase-space density f ( r , p ) is given by f ( r , p ) = 1 Z e − p / M + U ( r ) k B T Θ( − p / M − U ( r )) (A.2)with Z the partition function chosen such that R f ( r , p ) d r d p (2 π ~ ) = 1 , and Θ the Heavysidestep function. An integration of f ( r , p ) along the imaging direction z yields a 2D-profile n sim2D ( x, y ) n sim2D ( x, y ) = Z dz Z f ( r , p ) d p (2 π ~ ) = Z dz e − U ( r ) /k B T Γ inc (cid:18) U ( r ) k B T , (cid:19) , (A.3)where Γ inc is the incomplete gamma function. We also calculate the density of states ρ ( ǫ ) (with − V < ǫ < ) ρ ( ǫ ) = Z d r d p (2 π ~ ) δ (cid:0) ǫ − p / M − U ( r ) (cid:1) = 1(2 π ) (cid:18) m ~ (cid:19) / Z d r p ǫ − U ( r ) . (A.4)This can be used to determine the number of atoms N C that have an energy between − V and − V / , N C = N (cid:0) − V ǫ − V / (cid:1) = n Λ Z V / dǫ ρ ( ǫ )e − ǫ/k B T , (A.5)with n the density in the center of the trap, and Λ dB = h/ √ πmk B T the thermalde Broglie wavelength. We take N C as an estimate of the number of atoms in thecentral region. Equations (A.4) and (A.5) are evaluated numerically using Monte-Carlointegration.We apply to the simulated profiles n sim2D the same fitting routine as used for theexperimental images. In figure A1, we compare the fit output with the parameters usedfor the simulation. As one can see, the number of atoms in the center is found to be veryclose to N C . This validates our method to estimate of the loading ratio α = N C /N tot with the result from the fit N / ( N + N + N ) . Note however that the procedure sys-tematically over-estimates the temperature in the arms by ∼
30 % . This is due to thegaussian shape of the trap that causes a radial density profile wider than the profilecreated by a harmonic trap with the same curvature. We checked that for a truncatedharmonic trap, the fitted temperature is equal to the temperature T obtained for thesimulated profile (equation (A.3)). roduction of Sodium Bose–Einstein condensates in an optical dimple trap References [1] I. Bloch, J. Dalibard, and W. Zwerger. Many-body physics with ultracold gases.
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Physical Review Letters ,98(6), 2007. roduction of Sodium Bose–Einstein condensates in an optical dimple trap ! ! ! " ! ! ! ! " ! x y z ◦ a r m arm 2 CDT dT ( a )( b ) C D T p o w e r [ W ] d T p o w e r [ W ] t [s]t [s] Figure 1. (a) Sketch of the laser geometry showing the crossed dipole trap (CDT)propagating in the horizontal plane and the dimple trap (dT) propagating vertically.(b) Fluorescence image of the atoms trapped in the CDT taken after short time-of-flight . The thermal equilibrium state in such a potential has a characteristic spatialstructure: two elongated “arms” and a denser crossing region. (c) Evolution of thepowers of the crossed dipole trap and the dimple trap during the sequence. The firststep corresponds to the loading of the CDT from a “cold-MOT” phase, followed by acompression that helps to fill the central trapping region. The next step consists inevaporatively cooling the CDT and results in the filling of the dimple trap. Thelast step is evaporative cooling in the dimple trap, which leads to Bose–Einsteincondensation. roduction of Sodium Bose–Einstein condensates in an optical dimple trap ! !"" " !"" ! !"" ! ! !"" " !"" ! !"" ! ! !"" " !"" ! !"" ! ǫ = V / ǫ = V / ǫ = V × / !!!! V CDT (x , y , / !!!! x [ µm ] x [ µm ] x [ µm ] y [ µm ] y [ µm ] y [ µm ] ( a ) ( b ) ( c ) Figure 2.
CDT potential in the z = 0 plane, truncated at an energy ǫ = V / (a), ǫ = V / (b), and ǫ = V × / (c). Optimal loading N C Free evaporation time [s]Free evaporation time [s] ( a ) ( b ) α ∞ = 0 . , τ = 2 . s α ∞ = 0 . , τ = 5 . s α ∞ = 0 . , τ = 9 . s α ∞ = 0 . , τ = 4 . s α = N C / N . W . W W
2s ramp
Figure 3.
Evolution of the atom number in the center of the CDT (a) and the loadingratio α = N C /N (b) in four different loading situations: low power (circles), highestpower (diamonds), ∼ / of maximum power (crosses), ramping up in s after loadingat low power (stars). The loading ratio α ( t ) is fitted with the function a (1 − e − t/τ ) + b .The results of the fit τ and α ∞ = a + b are indicated in (b). roduction of Sodium Bose–Einstein condensates in an optical dimple trap Power [W] Power [W] τ [ s ] α ∞ Constant power
50 ms ramp ( a ) ( b ) Figure 4. (a) Filling time τ and (b) center filling fraction α ∞ for CDT in two differentsituations: the solid curve shows the results of the experiments where the CDT laseris hold at any time at the same power. The dashed curve denotes the compressionexperiments where the power starts at P CDT = 13 . W in the “cold MOT” phase andis ramped up in ms to the final value indicated after switching off the molassesbeams. The error-bars correspond to confidence bounds on the fit coefficients τ and α ∞ . For P CDT = 13 . W, both curves should intersect as the experimentalsequence is the same. The observed difference indicates systematic variations betweendifferent experimental runs, probably due to dipole trap pointing fluctuations and totalatom number variations. The vertical dashed line corresponds to the optimal power P CDT = 13 . W for the “cold MOT” phase. roduction of Sodium Bose–Einstein condensates in an optical dimple trap ! !" & $! ’ $! $! ( ! !" ! $ $! ! $! $ $! % ! !" ! ’ $! ! % $! ! $! % ! !" N t o t a l D T e m p e r a t u r e [ µ K ] ( a ) ( b )( c ) ( d ) α d t [ s ] t [ s ] Figure 5.
Evaporative cooling trajectories in the combined trap (CDT and dT)(circles) and in the CDT alone (stars). We show the time evolution of atom number N (a), temperature T (b), dimple filling α d (c) and phase-space density D (d). roduction of Sodium Bose–Einstein condensates in an optical dimple trap ! !"! ()$! ! !"! $$ $! $% $! $& $! $’ $! $ ! !"! ! $! ! ’ $! ! & $! ! % $! ! $ $! ! ) ) ( a ) ( b )( c ) ( d ) N C n [ a t . c m − ] T fi t / T D k B T /V k B T /V center arms thermal equilibrium gaussian fit k B T /V k B T /V
Figure A1.