Optimal geometry for efficient loading of an optical dipole trap

Optimal geometry for e ient loading of an opti al dipole trap Andrzej Sz zepkowi z,1 Leszek Krzemie«,2 Adam Woj ie howski,2 Krzysztof Brzozowski,2 Mi hael Krüger,2, 3 Mi haª Zawada,4 Mar in Witkowski,5 Jerzy Za horowski,2 and Woj ie h Gawlik2 1 Institute of Experimental Physi s, University of Wro ªaw, Pla 2 Institute Maksa Borna 9, 50-204 Wro ªaw, Poland of Physi s, Jagiellonian University, Reymonta 4, 30-057 Kraków, Poland arXiv:0809.1999v2 [physics.atom-ph] 31 Dec 2008 3 Max-Plan k-Institute of Quantum Opti s, Hans-Kopfermann-Str. 1, 85748 Gar hing, Germany 4 Institute of Physi s, Ni olaus Coperni us University, Grudzi¡dzka 5, 87-100 Toru«, Poland 5 Institute of Physi s, University of Opole, Oleska 48, 45-052 Opole, Poland (Dated: January 5, 2009) One important fa tor whi h determines e ien y of loading old atoms into an opti al dipole trap from a magneto-opti al trap is the distan e between the trap enters. By studying this e ien y for various opti al trap depths (2110 mK) we nd that for optimum dipole trap loading, longitudinal displa ements up to 15 mm are ne essary. An explanation for this observation is presented and ompared with other work and a simple analyti al formula is derived for the optimum distan e between the trap enters. PACS numbers: 37.10.Gh, 37.10.De, 37.10.Vz Keywords: Opti al Dipole Trap, ODT, FORT, Magnetoopti al trap, MOT, trap loading, old atoms, laser ooling and traping I. INTRODUCTION Opti al dipole trapping is an established tool for storing, manipulating and studying old atomi gases [1, 2, 3℄. Opti al dipole traps (ODTs) helped to a hieve Bose-Einstein ondensate of several elements [4, 5, 6, 7, 8℄ as well as to make ultra-stable frequen y standards [9℄. ODTs are also widely used for studies of various phenomena at quantum degenera y, su h as the Mott insulator  superuid liquid transition [10℄. The simplest realization of an opti al dipole trap (ODT) is a single, tightly fo used Gaussian laser beam, tuned far below the atomi resonan e frequen y. The trapping potential is proportional to the light intensity whi h, for a single beam, results in a highly elongated trap shape with opti al potential minimum at the trapping beam fo us. The ODT needs to be loaded by a pre ooled gas, and a standard way of loading atoms into the ODT is to transfer them from a magneto-opti al trap (MOT). Kuppens et al. [11℄ dis uss ways of maximizing this transfer by optimizing various parameters: MOT light intensities and detunings, magneti eld gradient, the ODT depth, and alignment of the traps. In parti ular, they found that ontrolling the geometry of the overlap of the MOT and ODT beams allows for substantial improvement of the transfer e ien y. They found that the loading rate is optimum with a longitudinal displa ement between the enters of the ODT and the MOT, and that this optimum displa ement in reases with the ODT depth. Optimal displa ements between the ODT and MOT enters reported in Ref. [11℄ were about one-half of a MOT diameter (the depths of the studied ODTs were 16 mK). These observations were rather qualitative and not a ompanied by systemati quantitative analysis. Sin e the loading of the trap is a ru ial step in any experimental work involving ODT, the purpose of our work was to determine quantitatively the optimum displa ement between the ODT and MOT enters for maximum transfer of atoms in a wide range of the ODT depths. The trap depths were hanged by varying the ODT laser power from 50 to 660 mW and detuning from 0.43 to 1.72 nm. The experimental setup and pro edure are des ribed in the next Se tion, while the obtained results are presented in Se tion III, and in Se tion IV we ompare quantitatively our experimental results with the models des ribed by Kuppens et al. [11℄ and O'Hara et al. [12℄. II. EXPERIMENT We use a standard MOT setup with three pairs of orthogonal, retroree ted beams for trapping 87 Rb atoms. The trapping light, red detuned from the 52 S1/2 F = 2 → 52 P3/2 F ′ = 3 transition, has a total six-beam power of 15 mW. The repumping light (0.5 mW, tuned to the 52 S1/2 F = 1 → 52 P3/2 F ′ = 2 transition) is added to one pair of the trapping beams. The radius of the beams is 8 mm (1/e2 intensity). The resulting MOT radius is 0.66 mm (1/e2 density). After olle tion, the atoms are further ooled by in reasing the detuning of the MOT beams to 100 MHz and de reasing the repumper intensity (see Fig. 1). At the end of the MOT ooling stage the MOT temperature rea hes TMOT = 17 µK, as determined from free expansion of the released atom loud. 2 FIG. 1: The time sequen e of loading the ODT from the MOT. The dipole trap is formed by a fo used beam of a Ti:Sapphire laser. The ODT beam is rst expanded, then fo used by a 25- m fo al length a hromati lens to a waist of 12 µm. The orresponding Rayleigh length is zR = 0.54 mm. We apply the ODT beam with powers, PODT , of 50, 100, 200, 400, and 660 mW, and red detunings from the D2 line, ∆λODT , of 0.43, 0.86, and 1.72 nm. The resulting trap depths determined by the ratio PODT /∆λODT were U0 /kB = 2.1, 4.2, 8.4, 17, 33, 67, and 111 mK. For the trap-depth estimation, a possible ee t of the Rb D1 line has been negle ted as it does not ex eed a few per ent. The transfer of the Rb atoms from the MOT to the ODT pro eeded with the timing shown in Fig. 1; it is similar to the sequen e used by Kuppens et al. [11℄. For ea h ODT laser power and detuning, the ODT-fo us position was gradually varied relative to the MOT enter. The des ribed loading sequen e (Fig. 1) was repeated for ea h position of the traps and the number of atoms loaded into the ODT was monitored by integrating the uores en e image aptured by the CCD amera. III. RESULTS Figure 2 shows the results of the loading measurements performed for the trap depth of U0 /kB = 33 mK. The uores en e images of the dipole trap are presented in Fig. 2 (a) for several dierent positions z of the ODT relative to the MOT enter. Superimposed are the ontours of the original MOT (1/e2 density ontour) and the dipole trap (the equipotential line at U = −2.5kB TMOT ). The diused louds seen below the trap ontours are the atoms whi h were not aptured into the ODT, falling in the dire tion of gravity after MOT has been swit hed o. Figure 2 (b) presents the total uores en e from the ODT, proportional to the number of trapped atoms, versus the fo us position. It is evident that when the enters of MOT and ODT oin ide (ODT fo us position z = 0), the number of aptured atoms is over 5 times smaller than in the ase of displa ed ODT: z = ±10 mm. The observed asymmetry between positive and negative z positions is attributed to geometri al imperfe tions. We believe that for perfe t alignment of the laser beams and magneti eld, the I(z) dependen es in Fig. 2 (b) should be symmetri . The dieren e in heights of the two urves shown in Fig. 2 (b) is most likely aused by some drift of the MOT onditions. We have ondu ted the des ribed measurement for a range of ODT trap depths, U0 /kB , between 2.1 and 111 mK. For the trap depth of 2.1 mK, we obtain a single, broad loading maximum at z = 0. The width of this maximum (FWHM) is 9 mm. For in reasing ODT potential depth, the maximum broadens and eventually, at 8.4 mK, splits into two separate maxima su h as shown in Fig. 2 (b). With our xed beam waist, the optimal separation of MOT and ODT depends only on the ratio PODT /∆λODT , that is, it depends only on the ODT depth. We have veried this for trap depths of 8.4, 17, and 33 mK realized by dierent ODT laser powers and detunings. We have veried that optimum displa ements z orresponding to a given PODT /∆λODT ratio were onstant within the experimental error of a few per ent for various ombinations of PODT and ∆λODT whi h preserved the PODT /∆λODT ratio. Figure 3 presents the dependen e of the optimum displa ement on the trap depths. It is worth noting that in the ase of the deepest opti al traps realized with our maximum ODT beam power, this optimum separation rea hes 15 mm. This displa ement is over 10 times larger than the diameter of the MOT, whereas the optimum displa ements reported in Ref. [11℄ were only of the order of one-half of a MOT diameter. 3 FIG. 2: Loading of the 33-mK-deep dipole trap for dierent positions of the ODP laser fo us with respe t to the MOT enter. (a) Fluores en e images of atoms in the ODT, verti ally stret hed for better visualization of the ODT potential. The ellipti ontours mark the MOT positions (0.66 mm radius) before swit hing on the ODT. The elongated ontours represent the ODT equipotential surfa es, U = −2.5kB TMOT . (b) Total uores en e from the ODT, proportional to the number of atoms loaded into the ODT, versus the fo us position. Solid line: PODT = 200 mW, ∆λ = 0.43 nm; dashed line: PODT = 400 mW, ∆λ = 0.86 nm. IV. DISCUSSION The dipole potential depends on the lo al light intensity I(r) and detuning from the atomi transition δλ as UODT (r) = 2π 2 c3 Γ I(r), ω05 δλ (1) where Γ denotes transition linewidth [11℄. The intensity of a fo used Gaussian beam of power P is des ribed by I(r, z) =   r2 2P , exp −2 πw2 (z) w2 (z) (2) p 1 + (z/zR )2 (3) where (r, z) are the ylindri al oordinates, and: w(z) = w0 is the 1/e2 beam radius. The hara teristi dimension in the radial dire tion is the beam waist w0 , while in the axial dire tion it is the Rayleigh length zR = πw02 /λ. The resulting trapping potential is visualized in Fig. 4. The potential is harmoni near the fo us, with ellipsoidal equipotential surfa es. However, the more distant equipotential surfa es are qualitatively dierent from the inner ones, and a quire hara teristi peanut shape. From equations (1) and (2) one an dedu e that the equipotential surfa es hange their hara ter when U be omes smaller than U ′ = U0 /e, where U0 is the trap depth and e is the base of the natural logarithm. The equipotential surfa es U = const with |U | < |U ′ | have two lo ations zmax where 4 Optimum MOT-ODT displacement z @mmD 25 20 15 10 5 0 0 20 40 60 80 ODT depth @mKD 100 120 FIG. 3: Optimum displa ement of the opti al dipole trap relative to the magneto-opti al trap for maximum transfer of atoms versus the ODT depth. s aling parameter The bla k dots represent our experimental data. α = 2.5, The solid line is al ulated from Eq. (5) with the as des ribed in the text. The dashed line is the predi tion based on the model of O'Hara et al. with no free parameters [12℄. See also Se . IV. FIG. 4: The ross se tions of the equipotential surfa es of the ODT formed by a fo used Gaussian beam. r/w0 is the U = 0.1 U0 . along the beam in units of the Rayleigh length. Dashed lines mark the positions of their ±zmax for radial oordinate in units of the beam waist. z/zR is the distan e U0 is the trap depth. ir umferen e is maximized: zmax = ±zR We will now r 1 U0 − 1. e U (4) ompare our experimental data with the two models of the ODT loading des ribed in the literature [11, 12℄. Kuppens et al. [11℄ assume that the initial loading rate is proportional to the MOT atomi mean velo ity of atoms parameters to v̄ , the ee tive surfa e area of ODT, A, and the trapping probability, density Ptrap . nMOT , the The rst three ontribute to the ux of atoms into the ODT volume. When studying geometri al ee ts, it is important onsider the ee tive surfa e area the ODT equipotential surfa es A. As the Rayleigh length is mu h larger that the size of the MOT, the radii of hange insigni antly over the MOT region (see also Fig. 6(a) below). The ee tive surfa e area a ross whi h atoms are loaded into the ODT is then proportional to the lo al radius of the ODT. The ru ial question in this reasoning is: whi h of the equipotential surfa es should represent the surfa e of the ODT. 5 Kuppens et al. assume the surfa e U ≈ kB TMOT , U where the ODT potential is omparable to the kineti energy of atoms in the MOT. In order to ompare quantitatively our data with the model des ribed above, we write the equation des ribing the ODT surfa e as U = αkB TMOT and look for a s aling fa tor α, of the order of unity, whi h gives the best t to our data. Equation (4) now takes the form z = ±zR r 1 U0 − 1. e αkB TMOT (5) Using the least-squares method, we nd that Eq. (5) best des ribes our experiment for α = 2.5. The result of this tting is shown by the solid line in Fig. 3. It turns out that despite its simpli ity, the model by Kuppens et al. gives a good qualitative des ription of our data, but needs the s aling fa tor α to be determined from the experiment. Another related work is that of O'Hara et al. [12℄ who developed a mi ros opi Fokker-Plan k equation. Their model is during ODT loading. For of atoms N (0) w is the ontained in the trap ([12℄, Eqs. 11, 12) 1/e2 for the loud, respe tively, and Z 1 dv v(− ln v) exp(−qv), ase when MOT and ODT enters U0 = min{U (r, z)}, r,z enters are displa ed axially. Then but as a lo al trap depth for a parti ular axial z: r U0 1 + (z/zR )2 ( ompare Eqs. (1) and (2)). On the other hand, the lo al beam radius moves away from the fo us, a shallower and more energeti w(z) (7) in reases from its initial value ording to Eq. (3). Consequently, as the axial oordinate z w0 as one in reases, the trap be omes atoms remain untrapped. On the other hand, the trap also be omes broader and atoms aptured from a larger volume. The balan e of these two for a R and n0 are the radius and al ulations of O'Hara et al. were performed The oin ide, but remain valid also if the U0 (z) = min{U (r, z)} = are (6) 0 q = U0 /kB TMOT . in Eq. (6) is treated not as a global trap depth position w2 2 q 2 beam radius (the width of the ODT potential in the radial dire tion), the density of the MOT U0 hanges of the atom density omparison with our results, the relevant equation is the expression for the initial number N (0) = (π 3/2 Rn0 ) where model of ODT loading based on the apable of des ribing the spatial and temporal ertain displa ement between the MOT and ODT The initial number of trapped atoms has been ompeting me hanisms results in a maximum loading enters. al ulated a ording to Eq. (6) for dierent displa ements z, treating w and q as z -dependent. The resulting fun tion is plotted in Fig. 5 for the trap depth U0 /kB = 33 mK. Note the maximum at z = ±13 mm, lose to our experimental result: z = ±10 mm (see Fig.3). Similarly, we al ulated the maxima of the N (z) fun tion for dierent trap depths and the result is shown by the dashed line in Fig. 3. It agrees the parameters qualitatively with the experimental data, but for deeper traps the predi ted optimum displa ement is overestimated. Nevertheless, the agreement is remarkable for a model that does not require any free parameters. It should be stressed CO2 laser traps, where the trap-indu ed ase of our experiment, hen e this may be that the model des ribed by O'Hara et al. was developed for lithium atoms in light shift of the atomi transition frequen y is small [12℄. This is not the the main reason of the dis repan y between our data and the model of O'Hara et al. Under onditions of our experiment, the frequen y shifts for the MOT trapping and repumping transitions may be taken with good a ura y as 2U (r, z)/h, where U (r, z) is the ODT potential whi h takes into a ount solely the D2 5 onditions, the P3/2 ex ited state makes the most 5 5 ontribution to the ODT potential and light shifts. Other states, P1/2 and D3/2, 5/2 , ontribute nearly line. As it was already mentioned above, under our experimental important two orders of magnitude less and an be negle ted at the urrent level of experimental pre ision. The light shift also modies the trapping transition but, given the large detuning (100 MHz) of the MOT beams in the loading phase, its ee t on the trapping-beam detuning is negligible. On the other hand, sin e the repumper beam is resonant, the repumping e ien y at a z = zmax , r = 0 is redu ed by about a fa tor of four (for all trap depths) F = 1 ground state. In Fig. 6, we ompare the spatial ranges in whi umulate atoms in the dark generates the repumper detuning with the spatial MOT extension seen in Figure 6(b), this redu tion ae ts only the whi h allows to h the light shift al ulated for the trap depth of 33 mK. As entral part of the MOT whi h, an be onsequently, results in a dark spot ee t whi h is known to in rease the MOT density [13℄ and enhan e the trap loading pro ess [14℄. Figure 6( ) shows that the light-shift-indu ed detuning of the repumper beam, and for in reasing z. For z = 13, onsequently the dark spot ee t, de rease whi h would be the optimum displa ement predi ted from the model of O'Hara et al., the repumper detuning amounts only 2.5 MHz and the dark spot ee t is mu h weaker. Thus, the light shift and the related dark-spot ee t de rease the optimum MOTODT displa ement relative to the predi tions of Ref. [12℄, 6 FIG. 5: The initial number of trapped atoms, N , as a fun tion of the MOTODT displa ement, z , predi ted by the model of O'Hara et al. [12℄ for our experiment (trap depth = 33 mK). FIG. 6: The geometri al superposition of the trap potentials and the inuen e of the ODT beam on the MOT. (a) The geometry of the MOT (represented by the 1/e2 density outline) and the 33-mK-deep ODT (represented by the equipotential surfa e U = −2.5kB TMOT  see text). The displa ement of the traps is optimized for maximum atom transfer. (b) The detuning of the MOT-repumper aused by the presen e of the ODT beam, plotted along the dashed line in (a). The gray outline represents the atomi density distribution of the MOT. ( ) Same as (b), but along the ODT beam axis (r = 0). Note the verti al s ale has been expanded for better visualization. as illustrated in Figure 3. We believe this is the main sour e of dis repan y between our results and the model of O'Hara et al. whi h refers to the ase of negligible light-shifts. Another fa tor of potential importan e for the ODT dynami s is the photon s attering rate. With a trap relatively lose to resonan e, su h as ours, the spontaneous photon s attering leads to atom heating and, eventually, to atom loss. We have estimated the heating rates under onditions of our experiment. For the deepest trap of 111 mK the s attering rate is 430 kHz and the heating during the 50 ms holding time is estimated to be about 2 mK. This estimation was done for the trap enter, whereas, as we demonstrated, the atoms are most e iently trapped far o 7 the enter (15mm), where the light intensity is signi antly lower and the lo al s attering rate is only 530 Hz. Thus, we believe that it is safe to negle t this me hanism in the ase of our experiment. V. SUMMARY We have performed a systemati study of the e ien y of the ODT loading from a MOT as a fun tion of geometri al arrangement of the ODT and MOT in a wide range of ODT depths (2 to 110 mK). We have observed that for the optimum loading the trap enters need to be displa ed in the dire tion of the ODT beam. While similar on lusion was formulated earlier in Ref. [11℄, we were able to verify it in a more than 20-times wider range. We onrmed that the optimum displa ement depends only on the trap depth, i.e., on the ratio between the trap light power and its detuning. Analyzing the optimum displa ements, we were able to quantitatively ompare our results with the existing models of Kuppens et al. [11℄ and O'Hara et al. [12℄. Our data are well des ribed by the former model after tting of a s aling fa tor and agree qualitatively with the latter model without any free parameters. Based on the present work, we propose the following semi empiri al formula for the optimal trap displa ement: z = ±zR r U0 1 − 1. e 2.5kB TMOT (8) A knowledgments This work was partly supported by Polish Ministry of S ien e, grant no. NN202175835. A.S. a knowledges also the funding from the University of Wro ªaw, grant no. 2016/W/IFD/2005. [1℄ [2℄ [3℄ [4℄ [5℄ [6℄ [7℄ [8℄ [9℄ [10℄ [11℄ [12℄ [13℄ [14℄ S. Chu, J. Bjorkholm, A. Ashkin, and A. Cable, Phys.Rev. Lett. 57, 314 (1986). H. J. Met alf and P. van der Straten, Laser ooling and trapping (Springer, 1999). R. Grimm, M. Weidemüller, and Y. Ov hinnikov, Adv. At. Mol. Opt. Phys. 42, 95 (2000). M. Barrett, J. Sauer, and M. Chapman, Phys. Rev. Lett. 87, 010404 (2001). R. Dumke, M. Johanning, E. Gomez, J. Weinstein, K. Jones, and P. Lett, New J. Phys. 8, 64 (2006). Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura, T. Yabuzaki, and Y. Takahashi, Phys. Rev. Lett. 91, 040404 (2003). A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Phys. Rev. Lett. 94, 160401 (2005). T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm, S ien e 299, 232 (2003). M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, Nature 435, 321 (2005). M. Greiner, O. Mandel, T. Esslinger, T. W. Häns h, and I. Blo h, Nature 415, 39 (2002). S. Kuppens, K. Corwin, K. Miller, T. Chupp, and C. Wieman, Phys. Rev. A 62, 013406 (2000). K. O'Hara, S. Granade, M. Gehm, and J. Thomas, Phys. Rev. A 63, 043403 (2001). W. Ketterle, K. B. Davis, M. A. Joe, A. Martin, and D. E. Prit hard, Phys.Rev. Lett. 70, 2253 (1993). S. Dürr, K. W. Miller, and C. E. Wieman, Phys. Rev. A 63, 011401 (2000).