Trapping and cooling of rf-dressed atoms in a quadrupole magnetic field
Olivier Morizot, Carlos L. Garrido Alzar, Paul-Eric Pottie, Vincent Lorent, Hélène Perrin
aa r X i v : . [ phy s i c s . a t o m - ph ] S e p Trapping and cooling of rf-dressed atoms in a quadrupole magnetic field
O. Morizot, C. L. Garrido Alzar, P.-E. Pottie, V. Lorent, and H. Perrin ∗ Laboratoire de physique des lasers, Institut Galil´ee, Universit´e Paris 13 and CNRS,Avenue J.-B. Cl´ement, F-93430 Villetaneuse, France (Dated: October 31, 2018)We observe the spontaneous evaporation of atoms confined in a bubble-like rf-dressed trap [1].The atoms are confined in a quadrupole magnetic trap and are dressed by a linearly polarized rffield. The evaporation is related to the presence of holes in the trap, at the positions where the rfcoupling vanishes, due to its vectorial character. The final temperature results from a competitionbetween residual heating and evaporation efficiency, which is controlled via the height of the holeswith respect to the bottom of the trap. The experimental data are modeled by a Monte-Carlosimulation predicting a small increase in phase space density limited by the heating rate. Thisincrease was within the phase space density determination uncertainty of the experiment.
PACS numbers: 32.80.Lg, 03.75.-b, 32.80.Pj
I. INTRODUCTION
Ultracold atoms in versatile traps is a subject ofextensive studies. Developing alternative trap geome-tries enables the exploration of new physical situations.Amongst the most remarkable results obtained withatoms in non harmonic traps, one may cite the observa-tion of superfluid to Mott insulator transition in opticallattices [2], Josephson oscillations in a double well [3]or trapping in a ring geometry [4]. Recently, radio-frequency (rf) fields were used together with a static mag-netic field to produce a quasi 2D trap [1, 5, 6] and a dou-ble well [7]. Other trapping geometries were proposed,based on this promising rf-dressing technique, such asrings [8, 9] or lattices [10]. One presents here the im-plementation of rf-dressing inside a quadrupole magnetictrap.Quadrupole traps are usually not popular for ultra-cold atom trapping due to Majorana losses at the centre,where the field vanishes. The losses are avoided in TOPtraps by adding a rotating uniform magnetic field, at afrequency in the 10 kHz range [11]. In the trap presentedin this paper, the magnetic field oscillates at a few MHz,resulting in an rf-dressed quadrupole trap. The atomsare then located away from the region of zero magneticfield. Still, the rf polarization is responsible for the ex-istence of regions of zero rf coupling. This results in aleakage of the higher kinetic atomic population. In thisLetter, we give evidence for a temperature decrease and anon exponential atom number loss, characteristic for anevaporation process, for different experimental parame-ters. ∗ Electronic address: [email protected]
II. TRAP DESCRIPTION
The trap relies on rf coupling between Zeeman sub-levels in an inhomogeneous static magnetic field. Thebasic idea of the rf-dressed potentials was first proposedby Zobay and Garraway [1], and experimentally demon-strated a few years later [5]. We will recall here the mainfeatures of this trap and develop further the consequencesof an inhomogeneous rf coupling.A static trapping magnetic field B ( r ) e B ( r ) present-ing a local minimum B is used together with a rf os-cillating magnetic field B rf cos( ω rf t ) e y . The rf field isdetuned by ∆ = ω rf − ω from the coupling frequency ω = g L µ B B / ¯ h at the static magnetic field minimum,where g L is the Land´e factor and µ B the Bohr magne-ton. The linearly polarized rf field couples the Zeemansubstates | F, m F i and | F, m ′ F i with m ′ F = m F ± m F levels into adiabatic states whoseenergy are represented on fig. 1. In the following, we willconcentrate on the upper adiabatic state, undergoing apotential V d ( r ) = F (cid:0) ( V B ( r ) − ¯ hω rf ) + ¯ h Ω ( r ) (cid:1) / . (1)This expression is obtained within the rotating wave ap-proximation. Here, m F = F = 2 in the case of Rb in its5 s / ground state. V B ( r ) = g L µ B B ( r ) is the magneticenergy shift between two adjacent Zeeman sublevels, andΩ( r ) = Ω sin( e y , e B ( r )) (2)is the Rabi frequency of rf coupling, of maximal ampli-tude Ω = g L µ B B rf / (2¯ h ). Ω is independent of r to avery good approximation. As deduced from this formula,the rf coupling vanishes at the points where the staticfield is parallel to the radio frequency polarization e y ,that is along the y axis. We will see later on that in thecase of a quadrupole static magnetic field, these pointsappear as two “holes” on the equator of our bubble-liketrap. -380(cid:13) -360(cid:13) -340(cid:13) -320(cid:13) -300(cid:13) -280(cid:13) -260(cid:13) -240(cid:13) -220(cid:13) -200(cid:13) -180(cid:13)-100(cid:13)-75(cid:13)-50(cid:13)-25(cid:13)0(cid:13)25(cid:13)50(cid:13) m(cid:13) F(cid:13) = -1(cid:13) m(cid:13)
F(cid:13) = 2(cid:13) E / h (cid:13) ( k H z ) (cid:13) z(cid:13) ((cid:13) m(cid:13) m)(cid:13) FIG. 1: Energy of the dressed levels of Rb in the magneticquadrupole trap described in the paper, plotted along thevertical coordinate z , in the vicinity of the potential minimumat z = − r . The parameters are ω rf = 2 π × . b ′ =150 G/cm and Ω = 2 π ×
100 kHz. The five dressed sublevelsfor a F = 2 spin state are plotted, as well as two bare statesfor comparison, m F = −
1, dashed, and m F = 2, dash-dotted.The atoms are trapped in the upper dressed potential m F = F = 2. Gravity was taken into account. The simplest situation occurs when the rf couplingΩ( r ) is uniform. Then, for a positive detuning ∆ > V d ( r ) is minimum all over an iso- B surface, defined by V B ( r ) = g L µ B B ( r ) = ¯ hω rf . Thisis no longer true if Ω varies in space, due to the spa-tial dependence of the orientation of the static field: thisbreaks the invariance on the iso- B surface, leading to afinite number of minima related to the minima of Ω( r ).On the other hand, gravity tends to push the atoms atthe bottom of the iso- B surface. To describe correctlythe potential geometry, we thus need to take both effectsinto account. As we will see, the potential minima willnot always sit exactly on the iso- B surface, but they willbe close to it in most relevant experimental cases.We concentrate in the following on the case of aquadrupole trap with an horizontal symmetry axis. Thisaxis is labeled y in our experimental setup, see fig. 2,and is the same as the rf polarization. Let b ′ be the mag-netic field gradient in the radial direction of coordinate ρ = ( x + z ) / , and let us define α as α = g L µ B b ′ / ¯ h .One has B ( r ) = b ′ ( ρ e ρ − y e y ) and therefore Ω( r ) =Ω ρ/ p ρ + 4 y . In this situation, an atom in the upper-most dressed state m F = F undergoes the potential V tot ( x, y, z ) = mgz (3)+ F ¯ h s ( α p ρ + 4 y − ω rf ) + ρ ρ + 4 y Ω where m is the atomic mass and mgz the gravitationalpotential. The relevant iso- B surface is then an ellip-soid of equation ρ + 4 y = r , where r is related to ω rf through r = ω rf /α . The rf coupling vanishes alongthe y axis, where the magnetic field is parallel to the rfpolarization.To visualize the potential, let us first remark that the FIG. 2: Simplified scheme of the trap. After accomplishingevaporative cooling in the QUIC trap (Quad + Ioffe coils)centered at position A , the atoms are loaded into the dressedrf trap (Quad + Ioffe + rf coils). The current in the Ioffe coilis then slowly ramped down to zero for transferring the atomsinto the dressed trap based on the quadrupole field of axis y (Quad + rf coils). The rf field polarization is aligned alongthe axis of the quadrupole. extrema are located in the vertical plane x = 0. In thisplane, ρ = | z | . One can rewrite V tot ( y, z ) in units of mgr as: V tot ( x = 0 , y, z ) / ( mgr ) = (4) zr + β vuut p z + 4 y r − ! + z z + 4 y Ω ω . The parameter β = F ¯ hα/mg is the ratio of the magneticand the gravitational force. It should be larger than 1for the magnetic trap to compensate gravity and in ourexperiment β ≃
10. Therefore, the behaviour of the sec-ond term in the potential expression is dominant. Indeed,the exact study of the minima position of V tot shows thanfor β ≫
1, the minima lie very close to the iso-B surface ρ + 4 y = r . To discuss qualitatively the potential ge-ometry, we will thus assume that the atoms sit exactlyon this ellipsoid, which cancels the first term under thesquare root. We are left with: V tot ( z ) / ( mgr ) ≃ zr + β | z | r Ω ω rf , with | z | ≤ r . (5)Depending on the ratio ω rf / Ω of the rf frequency tothe rf coupling, two different situations may occur: for ω rf / Ω < β , or equivalently mgr < F ¯ h Ω , the couplinginhomogeneity is the dominant effect, and the potentialminima sit at the points where the coupling term is thesmallest. This corresponds to the positions where ρ = 0,that is at the intersections of the iso-B ellipsoid and the y axis (see fig. 3 a ). Note that at these points, the rf cou-pling vanishes, leading to spin flip losses into untrappedstates. In the following, we will refer to these two pointsas to the “holes”. On the other hand, for ω rf / Ω > β , or mgr > F ¯ h Ω , gravity dominates and there is a singlepotential minimum at the very bottom of the ellipsoid,far from the holes (fig. 3 c ). In this single well, and in - 0.2 - 0.1 0 0.1 0.2y (a) (mm)- 0.3- 0.2- 0.100.10.2 z ( mm ) - 0.2 - 0.1 0 0.1 0.2y (b) (mm)- 0.3- 0.2- 0.100.10.2 z ( mm ) - 0.2 - 0.1 0 0.1 0.2y (c) (mm)- 0.3- 0.2- 0.100.10.2 z ( mm ) FIG. 3: Contour plot of the potential for Rb atoms in the yz plane, for different values of the rf coupling relative to ω rf .The parameters are ω rf / π = 3 . z direction, such that β = 9 .
85 and ω rf /β = 2 π ×
305 kHz. ( a ) large coupling ω rf / Ω < β (Ω / π = 1 MHz); the two minima are located at the position of theholes. ( b ) intermediate coupling ω rf / Ω = β (Ω / π = 305 kHz). ( c ) small coupling ω rf / Ω > β (Ω / π = 100 kHz); a singleminimum is present, at the bottom of the shell. the harmonic approximation, the oscillation frequenciesup to first order in 1 /β [12] are: ω x = r gr . (6) ω y = 2 r gr s − F ¯ h Ω mgr = 2 ω x r − β Ω ω rf . (7) ω z = α r F ¯ hm Ω = ω x r β ω rf Ω . (8)The vertical oscillation frequency was derived in [1]. Oneeasily identifies the pendulum frequency along x . Along y , the pendulum frequency is modified by the inhomo-geneous coupling. We remark that these oscillation fre-quencies are valid only for small amplitude oscillationsaround the potential minimum, the trap as a whole be-ing far from harmonic.For values of ω rf / Ω close to β , one will face a situ-ation where the potential is almost flat in the crescentdefined by z < x = 0 and z + 4 y = r (fig. 3 b ). Thegravitational energy mgr at the holes is indeed exactlyequal to the energy shift F ¯ h Ω induced by the rf at thebottom of the iso- B surface, and ω y as expressed abovegoes to zero. However, this intermediate region is ex-tremely narrow for β ≫
1, with a width in the parameter ω rf / Ω on the order of 3 / (2 β ) around β , and in practiceone deals only with one of the two extreme situations, a or c . Note that in any case the zero magnetic fieldregion, responsible for Majorana spin flips in the centreof conventional quadrupole traps, plays a minor role inan rf-dressed quadrupole trap. The energy barrier is atleast F ¯ hω rf , much larger than the thermal energy of theatomic cloud.In our experiment, ω rf / Ω is larger than β (typically ω rf / π ≃ / π <
260 kHz), and we arealways in the case of a single minimum. Therefore, dueto gravity, ultracold atoms remain trapped near the bot-tom of the ellipsoid and only marginally explore the non-coupling regions. This single-well situation was demon- strated experimentally in a Ioffe-Pritchard type trap [5].The opposite situation was used in an atom chip exper-iment for producing a double well potential [7]. In theexperiment described in this paper, we bring into lightthe non negligible effect of the two holes for an ultra-cold atomic cloud, even in the situation where a singleminimum is present.
III. EXPERIMENTAL SEQUENCE
The experimental setup has been described in detailelsewhere [13, 14]. The atoms in the state F = 2, m F = +2 are first loaded into a static magnetic trap, ina Quadrupole and Ioffe Configuration (QUIC) [15], seefig. 2, Quad and Ioffe coils. In this trap, the atoms areconfined near the magnetic field minimum, position A offig. 2, situated 7 mm away from the quadrupole center O in the direction of the Ioffe coil. Then, a 30 s radio fre-quency ramp is applied to evaporatively cool them downto just above the condensation threshold. We deliber-ately do not reach condensation and rather work withultracold clouds, with a temperature of typically 4 µ K,in order to explore the leakage through the pierced partof the iso-B bubble as explained later. The evapora-tion rf source is then switched off and the trapping rffield, polarized along y , is turned on at a frequency lowerthan the resonance frequency in the trap centre, 1.3 MHz.Note that for this purpose, a non zero minimum magnetictrap is necessary. The rf frequency is then adiabaticallyramped up to the desired value of ω rf within typically150 ms, as detailed in [5]. This way, the atoms are al-ways following the upper dressed state, corresponding to m F = F = +2 in the dressed basis. At the end of thisstep, they are confined in a dressed rf trap relying on theQUIC static magnetic field.The QUIC trap stage is necessary to obtain an ultra-cold sample and load it efficiently into the dressed rf trap.The atoms are then transferred into a dressed quadrupoletrap, which presents points of strictly zero rf coupling atany value of ω rf . The current in the Ioffe coil is thereforedecreased in order to reach a quadrupole configurationwithin a time interval of about 450 ms. During this pro-cess, the minimum of the static magnetic field quicklygoes to zero and is split into two zero field points sepa-rating along the x axis. One point is going back 7 mmaway from A to the position O of the initial quadrupolecentre whereas the other one is going from A to infinityin the opposite direction (see Ref. [15], Fig. 2). Duringthis transfer, the atomic cloud remains in the dressedtrap, situated just below the static magnetic minimum,and is split into two clouds according to the deformationof the iso-magnetic surfaces. The first stage of the Ioffecurrent decreasing is controlled carefully in order that atleast half of the atoms follow the right direction, towardthe initial quadrupole centre O . At the end of this stage,about 4 × atoms at a higher temperature of 8 µ K areconfined in an rf-dressed quadrupole trap. The QUIC-to-quadrupole transfer of the rf-dressed atoms is a criticalstep and is very sensitive to initial conditions. This pro-cedure is at the origin of a scattering in the values ofatom number and temperature.The rf field is produced by a home made synthe-sizer [16]. The Rabi frequency Ω corresponding to agiven voltage amplitude is evaluated as follows. The rfantenna, normally dedicated to rf trapping, is used forevaporative cooling in the QUIC trap for this measure-ment. An evaporation ramp is applied down to a givenfinal rf frequency. The threshold frequency below whichall atoms are expelled from the magnetic trap is recorded.For an arbitrary weak rf power, this threshold frequencycorresponds to the Zeeman splitting ω at the bottom ofthe QUIC magnetic trap. However, the recorded thresh-old frequency is larger, as the magnetic levels are de-formed due to dressing by the rf photons. We use theobserved value as input in the calculation of the dressedpotential, including gravity, and search for the Rabi fre-quency value making the potential flat, just unable tohold the atoms. We repeat this procedure for differentvoltage amplitudes at the synthesizer to calibrate the rfpower. IV. EXPERIMENTAL RESULTS
After the atoms are transferred to the rf-dressedquadrupole trap, they remain stored for a variable time τ after which the magnetic field and rf coupling are si-multaneously switched off. An absorption image of thecloud is taken after a 7.5 ms ballistic expansion. Fromthese measurements, the evolution of the atom numberand the temperature with the storage time τ is deduced.The temperature is related to the vertical size after ex-pansion of the atomic cloud. In principle, it also dependson the initial vertical size, which is not entirely negligibledue to the curved shape of the trap. However, we neglectthe initial size, which overestimates the real temperatureby about 20%, as deduced from the comparison with a t e m pe r a t u r e ( µ K ) (cid:13) holding time (s)(cid:13) FIG. 4: Temperature T , measured as described in the text,as a function of time with Ω = 2 π ×
40 kHz and for differentvalues of the RF frequency ω rf / π = 2 MHz (closed circles)and 3 . T drops over the first trapping seconds, and then stabilizes to avalue T = 1 . µ K (resp. 3 . µ K) indicated by the horizontallines, deduced from the average of the 5 (resp. 4) last points. numerical simulation, see next section.We observe a rapid decreasing of the temperature dur-ing the first few seconds. The experiment was repeatedat Ω / π = 40 kHz for two different values of ω rf (seefig. 4). For ω rf / π = 2 MHz, we find a final tempera-ture T = 1 . µ K whereas it does not get lower than T = 3 . µ K for ω rf / π = 3 . / π = 260 kHz, the temperature de-crease, can be observed over more than ten seconds (seethe experimental points on fig. 5), with a first drop intemperature followed by a transition to a slower but stilldecreasing evolution.These results, together with the non exponential de-crease of the atom number, see inset of fig. 5, reveal anevaporation process via Landau-Zener losses through theholes along the rf coil axis in the equatorial plane. Thefinal temperature is related to the trap depth, calculatedbetween the trap bottom and the position of the holesand depending on the rf frequency through U = mgr − F ¯ h Ω = mgω rf α − F ¯ h Ω . (9)The trap depth is U /k B = Θ = 26 . µ K (resp.Θ = 15 . µ K) for a 3.1 MHz (resp. 2 MHz) rf frequencyand a Rabi frequency Ω / π = 40 kHz. The final tem-perature compared to the trap depth gives Θ /T = 7 . /T = 11 .
6. The overestimation of temperaturemakes these values a lower bound to the depth to tem-perature ratio. A ratio of 8 to 10 is considered typicalfor a spontaneous evaporation process in an harmonictrap [17].In principle, the evaporation could also come fromlosses through the centre of the quadrupole, however theenergy F ¯ hαr = F ¯ hω rf required to cross the central re-gion of the quadrupole trap is much higher than the en-ergy U necessary to explore the uncoupled region, mak-ing this process very unlikely. T ( µ K ) (cid:13) t (s)(cid:13) experiment(cid:13) simulation(cid:13) experiment(cid:13) simulation(cid:13) exponential fit(cid:13) (cid:13) (cid:13) N ( (cid:13) (cid:13) ) (cid:13) t (s)(cid:13) FIG. 5: Temperature, deduced from the size σ z after a 7.5 mstime-of-flight, as a function of the holding time in the rf-dressed quadrupole trap. Experimental data, closed dia-monds with error bars as in Fig. 4, are compared to the nu-merical simulation, thick line. Inset: Evolution of the atomnumber in the same conditions: experimental data, circles,are compared to the simulation, line and to a pure exponen-tial fit, dashed red line. For these experiments and the re-lated simulation, the parameters are ω rf / π = 3 . = 2 π ×
260 kHz.
Finally, the possible non-adiabatic Landau-Zenerlosses in the avoided crossing region at the bottom of theellipsoid, where the atoms mostly sit, can be neglected.Indeed, the probability of spin-flipping in this zone for agiven velocity v reads [18] P LZ = 1 − (cid:20) − exp (cid:18) − π αv (cid:19)(cid:21) F . (10)If one averages Eq. (10) over the whole velocity distri-bution for an rf coupling Ω = 30 kHz, smaller than wasever used in the experiment, and a temperature in thedressed trap T = 10 µ K, one finds a life time τ LZ due toLandau-Zener losses as high as 100 s, which is even largerthan the life time due to collisions with the backgroundgas. Landau-Zener losses at the trap bottom are thenobviously not responsible for the losses and temperaturedecrease. V. NUMERICAL SIMULATION
In order to confirm our interpretation, we performeda numerical simulation of the cloud classical dynamicswith the parameters of the experiment, for a couplingΩ / π = 260 kHz, a rf frequency ω rf / π = 3 . T i , above condensation threshold. Collisions betweentrapped atoms are taken into account through a methoddeveloped by G.A. Bird [19], while collisions with thebackground gas are described through a lifetime of 60 s.A possible linear heating rate H is taken into accountin the simulation by increasing each velocity component (cid:13) (cid:13) pha s e s pa c e den s i t y ( - (cid:13) ) (cid:13) t (s)(cid:13) experiment(cid:13) simulation(cid:13) FIG. 6: Estimated phase space density for the data seriesof Fig. 5, from the experimental data, diamonds, and thesimulation, line. The parameters are ω rf / π = 3 . = 2 π ×
260 kHz. The error bars take into account datadispersion and an experimental uncertainty of 40%, resultingfrom uncertainties in atom number and temperature. v i by 2 v i Hdt/T after a time step dt . The Landau-Zenerlosses are more difficult to model. The use of a randomnumber to decide, with a local probability distribution, ifan atom is lost or not would be time-consuming. Instead,the holes are modeled by a deterministic loss term: theatoms are considered lost if the Landau-Zener probabilityat this point P LZ ( r , v ) exceeds a critical value P c . Dueto the exponential dependence of the Landau-Zener nonadiabatic coupling on the rf Rabi frequency, this occursessentially in a region very close to the two holes.Fig. 5 gives a plot of the temperature for different hold-ing times in the dressed trap. The temperature is de-duced in the same conditions as in the experiment, fromthe cloud size after a 7.5 ms time-of-flight, and the twovalues are thus directly comparable. The simulation fitsat best the experimental data for the choice P c = 0 . H = 300 nK/s. The Landau-Zener probability P c mainlydetermines the initial rate of temperature decrease. Oncethis value is fixed, the heating rate H imposes the finalsaturation temperature, resulting from the equilibriumbetween evaporation and residual heating. The simula-tion reproduces well the features of spontaneous evapora-tion, namely temperature reduction and non exponentialatomic losses, see dotted line in the inset of fig. 5. Thesefeatures completely disappear if the holes are suppressed.The peak phase space density ϕ in the rf-dressed trap iscomputed from the simulated data. To be able to com-pare this figure with the experiment, we used in bothcases the estimation ϕ ≃ N (¯ h ¯ ω/k B T ) , where T is cal-culated from the vertical width after time-of-flight as ex-plained above. Here, ¯ ω is the geometric mean of the os-cillation frequencies of the rf-dressed trap, Eq.(6-8). Thisestimate only gives the order of magnitude for the phasespace density, as the formula is correct only for a har-monic trap and the trap is far from being harmonic. Theresults are displayed on Fig. 6, for the same parametersas used in Fig. 5. For these parameters, ¯ ω/ π = 60 . ϕ of the same order, but are too scattered fordemonstrating unambiguously a phase space density in-crease. This is due to the sensitivity of the estimate for ϕ to the temperature value T , which is determined witha large uncertainty. VI. CONCLUSION
In this paper, we demonstrate trapping of ultracoldatoms in a quadrupole magnetic configuration. Atomsare prevented to escape through the centre by rf-dressingof the atomic state, which results in a confining adia-batic potential. Non exponential losses associated to atemperature decrease are observed in this trap, givingevidence for spontaneous evaporation. These losses areattributed to the presence of zero rf-coupling zones in thetrap, due to rf polarization. Experimental data are inter-preted with the help of a numerical simulation, well re-producing the main features. The evaporation results ina drop in temperature, which saturates due to a residualheating in the trap. This technical heating clearly pre-vents an increase in phase space density. Cooling in the trap may be improved in different ways. The observedspontaneous evaporation could be forced by reducing therf frequency ω rf with time, thus limiting the trap depth.Still, it seems that a necessary approach would be touse a second independent rf field to spin flip selectivelythe more energetic atoms, as in conventional evaporativecooling [20, 21]. Ultimately, the quadrupole geometrywith a symmetry axis oriented in the vertical direction isideal for the realization of a isotropic quasi-2D degener-ate gas [22] or an atomic ring [8]. Acknowledgments
We acknowledge fruitful discussions with Barry Gar-raway. This work was supported by the R´egion Ile-de-France (contract number E1213) and by the Euro-pean Community through the Research Training Net-work “FASTNet” under contract No. HPRN-CT-2002-00304 and Marie Curie Training Network “Atom Chips”under contract No. MRTN-CT-2003-505032. Labora-toire de physique des lasers is UMR 7538 of CNRS andParis 13 University. The LPL group is a member of theInstitut Francilien de Recherche sur les Atomes Froids. [1] Zobay O and Garraway B M 2001
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