Ultracold chromium atoms: From Feshbach resonances to a dipolar Bose-Einstein condensate

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1 Journal of Modern Optics Vol. 00, No. 00, DD Month 200x, 4 Ultracold chromium atoms: From Feshbach resonances to a dipolar Bose-Einstein condensate Jürgen Stuhler, Axel Griesmaier, Jörg Werner, Tobias Koch, Marco Fattori, Tilman Pfau 5. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, D Stuttgart, Germany (2005) We report on experiments with ultracold chromium atoms. Preparing a cloud of 52 Cr atoms in a crossed optical dipole trap (CODT) and applying magnetic fields between 4 and 600 G, we observe 4 Feshbach resonances by the occurrence of increased atom loss at distinct magnetic field values. A comparison with theory taking only dipole-dipole coupling into account shows very good agreement between experimental and theoretical resonance positions and allows us to extract the s-wave scattering lengths a 6 = 2(4) a 0, a 4 = 58(6) a 0, a 2 = 7(20) a 0 of the involved molecular potentials as well as the dispersion coefficients C 6 = 733(70) a.u. and C 8 = a.u.. The strongest resonance at 589 G has a calculated width of.7 G and reveals a three-body loss coefficient below L 3,max m 6 /s. Further evaporative cooling within the CODT leads to the formation of a Bose-Einstein condensate (BEC) with up to 00,000 condensed atoms. The magnetic dipole-dipole interaction between the Cr atoms is so strong that, as a first mechanical manifestation of dipole-dipole interaction, we observe a modification of the condensate expansion which depends on the alignment of the atomic magnetic dipoles with respect to the axis of the CODT. This magnetostrictive effect is in very good agreement with the theory of dipolar quantum gases and shows that a Cr-BEC is an excellent model system to study dipole-dipole interactions in degenerate quantum gases. Introduction Over the past decade, ultracold atomic samples have become workhorses of quantum optics, atomic physics and solid state physics. Especially degenerate quantum gases have attracted much attention. However, so far, mainly atoms with either one or two valence electrons have been studied. As a consequence, the van der Waals interaction, which is short-range and isotropic, was the dominant interaction mechanism present. At low temperatures, this interaction can be described by a contact potential with one significant parameter, the s-wave scattering length a. Many exciting phenomena based on this contactlike interaction have been studied and Bose-Einstein condensates (BECs) have been used as model systems for solid state physics problems (an overview can be found e.g. in [ 3]). Utilizing so-called Feshbach resonances [4] molecular

2 2 Stuhler, Griesmaier, Werner, Koch, Fattori, Pfau BECs [5 8] have been realized and the crossover from a molecular BEC to a degenerate Fermi gas has been studied [9 3] and even superfluidity has been observed in a strongly interacting Fermi gas [4]. Such a Feshbach resonance occurs when the potential curves of different but interaction-coupled molecular states of colliding atoms cross at certain values of external (at most magnetic) fields. At such field values, the scattering process between the particles is resonantly enhanced, leading to a modification of the s-wave scattering. The successful generation of a BEC of chromium atoms [5] provides access to a completely new class of experiments relying on dipole-dipole interaction. In contrast to the contact interaction, the dipole-dipole interaction is long-range and anisotropic. In the past few years, especially in the context of degenerate quantum gases, dipole-dipole interaction has attracted much interest [6]. Motivated by theoretical work, much experimental effort was undertaken to realize degenerate quantum gases with strong or even dominant dipole-dipole interaction. Most of the experiments dealt and are dealing with polar molecules since they provide large electric dipole moments and consequently show strong dipole-dipole interaction. Although much promising progress has been made (an overview can be found in [7]), a dipolar degenerate quantum gas of polar molecules has not been realized so far. Another possibility is, however, to use atoms with large magnetic moments, like chromium. In its electronic ground state, chromium atoms have a magnetic moment of 6 Bohr magnetons that is 6 times larger than the one of alkali atoms. Hence, the magnetic dipole-dipole interaction (MDDI) in chromium is a factor of 36 stronger than in alkali systems, enough to become comparable to the contact interaction and observable in experiments. For example, it was predicted that the expansion of a Bose-Einstein condensate (BEC) of chromium atoms should depend on the orientation of the magnetic moments [8]. Other generally discussed dipolar phenomena are e.g. the existence of a local minimum in the dispersion relation (Roton-Maxon) [9], the dependence of the stability and the shape of dipolar BECs on the shape of the trap and the orientation of the dipole moments [20 23], and even new quantum phase transitions [24]. In this article, we report on recent experiments with ultracold chromium atoms. Our starting point is a cloud of the most abundant isotope 52 Cr confined in an optical dipole trap and fully polarized in the energetically lowest lying Zeeman substate. The preparation of this sample will be shortly summarized in the next section. Applying a homogeneous magnetic field in the range of 4 to 600 G and detecting the number atoms that remain in the trap after a certain holding time, we observe 4 Feshbach resonances. The measurement procedure, the experimental results and their interpretation as well as their significance will be described in section 3. Subsequently, we will sketch the way to realize a BEC of chromium which becomes visible in absorption images of the expanding atomic cloud. In section 5, we report on the observation

3 Ultracold Cr: from Feshbach resonances to a dipolar BEC 3 of dipole-dipole interaction in the Cr-BEC. We show that the condensate expansion depends on the orientation of the magnetic moments of the chromium atoms which is the first mechanical manifestation of dipole-dipole interaction in a gas. We conclude with a short summary of the results and their scientific impact and give an outlook on future experiments. 2 Preparation of chromium atoms in a crossed optical dipole trap Several experimental steps are required to load chromium atoms into an optical dipole trap. They are described in detail in [25] and will be summarized briefly in the following. An atomic beam is produced by the sublimation of chromium within a high-temperature effusion cell operated at T = 600 C. Fast atoms are decelerated by a Zeeman slowing stage and continuously loaded into a cloverleaf-type Ioffe-Pritchard magnetic trap [26,27] where they are trapped in a metastable electronic state ( 5 D 4 ). After repumping the atoms into the ground state ( 7 S 3 ), an additional Doppler cooling stage is implemented to increase the phase space density by a factor of 80 [28]. A subsequent radio-frequency evaporation step becomes inefficient at a phase space densities below 0 4 [29] due to dipolar relaxation processes which are much stronger in chromium than e.g. in rubidium, due to its larger magnetic moment. To circumvent this loss mechanism, the atoms are transferred into a single beam optical dipole trap and optically pumped from the magnetically trapped Zeeman substate m s = 3 into the energetically lowest lying substate m s = 3. The optical trap is formed by a tightly focused beam of a fibre laser (IPG PYL-20M-LP, waist w 0 = 30 µm, wavelength λ = 064 nm, power P 9 W) aligned horizontally and collinearly to the weak axis of the magnetic trap. After or during a stage of plain evaporation, a second beam of the same laser (waist w 0 = 50 µm, P 6 W), which is aligned vertically through the centre of the horizontal laser beam, is ramped up slowly to form a crossed optical dipole trap (CODT). This is followed by a stage of forced evaporative cooling within the optical dipole trap achieved by ramping down the intensity of the horizontal laser beam. With decreasing intensity of the horizontal laser beam more and more atoms are confined in the dimple, the region where both laser beams overlap. Initial and final intensities of both laser beams as well as the number of steps and the time durations of the evaporation ramps depend on the experiment to be performed.

4 4 Stuhler, Griesmaier, Werner, Koch, Fattori, Pfau 3 Feshbach resonances in collisions of ultracold chromium To detect the Feshbach resonances, we analyze atom loss from the CODT as a function of the applied magnetic field. For this, we start with initial laser powers of W and 6 W in the horizontal and vertical optical dipole trap laser beam, respectively. Forced evaporation is continued by ramping down the horizontal laser beam to 5 W. This results in N 20, 000 Cratoms within the CODT at temperatures of T 6 µk and peak densities of n cm 3. The magnetic field B is generated either using a pair of coils in Helmholtz configuration (for 0 G < B = B < 20 G) or utilizing the pinch coils of the cloverleaf magnetic trap (for 20 G B < 600 G). In the first case, the magnetic field shows no significant variation over the size of the atomic cloud. In the second case of using the pinch coils, variations of the magnetic field on the order of 50 mg at the highest applied magnetic fields of 600 G cannot be ruled out and enter the error budget for the position and the width of the Feshbach resonances. Other error contributions are due to the finite temperature of the atomic sample and due to noise in the power supplies that drive electric current through the coils. The absolute value of the magnetic field is calibrated above and below each resonance by radio frequency (rf) spectroscopy. Depending on B, a certain rf has to be applied to the chromium cloud in order to induce transitions from the lowest lying Zeeman substate m s = 3 to m s = 2,,... thereby depositing the rf-energy in the spin reservoir of the cloud. Subsequent dipolar relaxation releases this energy into kinetic energy which is redistributed among the atoms. This causes heating of the cloud and leads to atom loss by evaporation. Figure shows the number of remaining atoms as a function of the applied radio frequency exemplarily for a magnetic field of B 97.2 G. From the resonance rf, one can evaluate the magnetic field by equating the rf energy with the Zeeman energy between to adjacent magnetic substates. For magnetic field values above 450 G, we extrapolate from the measurements performed at lower field strengths. The resulting accuracy in determining the magnetic field is better than 00 mg. Taking all error sources into account, the experimental resolution is limited to 40 mg for the highest lying Feshbach resonances and better for the lower ones. Near a Feshbach resonance, the s-wave scattering length shows a dispersive behaviour and the three-body loss rate increases [30] which leads to increased atom loss and heating. To locate the Feshbach resonance positions, we first perform slow ramp steps of the magnetic field ( 3 G in a few seconds) from 0 to 600 G. Regions where we observe increased atom loss are subsequently investigated with higher resolution. For this, we switch the magnetic field to values slightly above the resonance, let the current settle and the coils rethermalize. Then the magnetic field is ramped at maximum speed to specific mag-

5 Ultracold Cr: from Feshbach resonances to a dipolar BEC 5 Number of atoms [0 4 ] RF Frequency [MHz] B=97.9 G σ=86 mg Magnetic field [G] Figure. Calibration of the magnetic field. Remaining number of atoms as a function of the applied radio frequency and the magnetic field as evaluated from equating the rf energy with the Zeeman energy between to adjacent magnetic substates (e.g. m s = 3 and m s = 2). netic field values which are then kept for a certain holding time. Then the cloud is absorption imaged after several milliseconds of free expansion to get the necessary information. The holding time lies between 0. and 0 s and is chosen such that the resonance is clearly resolved but the atom loss is not saturated. Figure 2 shows the two characteristic features of a Feshbach resonance, increased atom loss and heating, as a function of the applied magnetic field exemplarily for the resonance around G. By a gaussian fit, which we used for convenience, to the data, we find the position of the resonance. All fourteen observed resonances are shown in figure 3. (a) Relative number of atoms (c) Magnetic field [G] (b) Temperature [µk] Magnetic field [G] T z T y Figure 2. Feshbach resonance at G. Shown are (a) the relative atom number and (b) the temperature over the magnetic field. (c) Corresponding absorption image pictures of the atomic cloud (every second picture has been omitted for space reasons).

6 6 Stuhler, Griesmaier, Werner, Koch, Fattori, Pfau M S =-3 S=6 S=4 S=2 `=2 `=4 `=2 `=4 M S =-2 M S =-4 M S = G, 0.47 G 50. G, 0.4 G 65. G, 0.09 G 98.9 G, 0.09 G G, 0.68 G G, 0.42 G `=0 4. G, 0.04 G 8.2 G, 0. G G, G G, 0.37 G G, 0.43 G G, 0.2 G G, 0.4 G not yet assigned: G, G increasing B-field atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] atom number [a.u.] Figure 3. Atom loss features, molecular bound state quantum numbers (S, M s = spin and its projection onto B, l = nuclear angular momentum) and, above each curve, magnetic field B exp and the experimental width σ exp of all observed Feshbach resonances (see table ).

7 Ultracold Cr: from Feshbach resonances to a dipolar BEC 7 The eleven strongest Feshbach resonances build a complete set. Starting from s-wave (l = 0, M l = 0 with nuclear angular momentum of the two atoms and its projection M l = 0 onto the quantization axis) open channel collisions of two m s = 3 atoms (S = 6, M s = 6, with total spin S and its projection M s = 6) and taking the selection rules into account, one can predict the number of Feshbach resonances. For dipole-dipole coupling between colliding atoms and bound molecular states, the selection rules read S = 0, ±2; l = 0, ±2; M l = 0, ±, ±2 () for first order coupling and for second order coupling one gets S = 0, ±2, ±4; l = 0, ±2, ±4; M l = 0, ±, ±2, ±3, ±4. (2) In both cases, conservation of angular momentum requires M s + M l = 0 (3) and transitions from l = 0 to l = 0 are forbidden. Additionally, M s 0 is required to allow for magnetically shifting the open channel (incoming colliding atoms) and the closed channel (bound molecular state) into resonance. Following figure 4 (red solid arrows) and taking all the rules mentioned above into account, one can see that the only states with M s 6, to which the S = 6, M s = 6, l = 0, M l = 0 state of the open channel is first order dipole-dipole coupled, are the states S = 6, M s = 5, l = 2, M l =, S = 6, M s = 4, l = 2, M l = 2 and S = 4, M s = 4, l = 2, M l = 2. Hence, three first order Feshbach resonances are predicted. Second order dipole-dipole coupling gives eight more possible resonances (blue dashed arrows), so in total one expects eleven Feshbach resonances. All of them are observed. Two of the weaker resonances that have been found in addition at low magnetic fields belong to d-wave entrance channel collisions. The resonance at 6.4 G remains unassigned. The results of multi-channel scattering calculations are compared to the experimental data [3] in table and in figure 5. In this calculations, the molecular potential is modeled by five adjustable parameters (a 6, a 4, a 2, C 6, C 8 ) and dipole-dipole coupling up to second order is considered. The best agreement between theory and experiment is obtained by a least square fitting procedure to optimize the molecular potential parameters and results in the following values (in Bohr radii a 0 and atomic units a.u., respectively): a 6 = 2(4) a 0, a 4 = 58(6) a 0, a 2 = 7(20) a 0, C 6 = 733(70) a.u. and C 8 = a.u.. The broadest Feshbach resonance at 589 G has a theoretical width of.7 G and comes along with a maximum three-body loss coefficient of L m 6 /s. This value is much smaller than the

8 8 Stuhler, Griesmaier, Werner, Koch, Fattori, Pfau Figure 4. Overview over expected Feshbach resonances up to second order dipole-dipole coupling for two colliding ground state Cr aroms (open channel S = 6, M s = 6, l = 0, M l = 0 ). Three first order and eight second order resonances are expected due to the selection rules (eqns. 3). (a) 5 0 a 6,-6 [a 0 ] (c) (b) Relative number of atoms Magnetic field [G] Figure 5. a) Calculated magnetic field dependence of the s-wave scattering length a S,MS for model parameters a 6 =.56 a 0, a 4 = 57.6 a 0, a 2 = 7.26 a 0, C 6 = 733 a.u., C 8 = a.u. b) Plot of the experimental loss features. c) The feature near 290 G consists of two resonances [3]. one observed in 85 Rb [32], smaller than the one for Cs [33] but larger than the ones observed in 23 Na [30] and 87 Rb [34]. It is expected that the Feshbach resonances in Cr can be used to tailor strength and sign of the contact interaction. In summary, for the first time, we have observed Feshbach resonances with

9 Ultracold Cr: from Feshbach resonances to a dipolar BEC 9 B exp [G] B theo [G] σ exp [mg] theo [mg] S, M S ; l, m l < 0 3 6, -2; 4, , -3; 4, , -4; 4, , -2; 4, , -3; 4, , -5; 4, , -4; 4, , -4; 2, , -2; 4, , -4; 2, , -5; 2, - Table. Compendium of positions B exp and / e-widths σ exp of the observed loss features, the theoretical positions B theo, widths theo, and assignment (only closed channel quantum numbers, open channel is always S = 6, M s = 6, l = 0, M l = 0 ) of the eleven strongest Feshbach resonances. Theoretical calculations use a collision energy of E = k B T and parameters as described in the text. The one standard deviation uncertainty of the experimental resonance position is below 00 mg. an element that has more than one valence electron. All of the eleven expected resonances have been detected. The positions of the resonances can be calculated to very high accuracy taking into account only dipole-dipole coupling which does not require an additional free parameter. This makes Cr a textbook element for Feshbach theory, despite of having six valence electrons. The resonances at 589 G and 290 G are promising candidates for Feshbach tuning of the scattering length a 6, 6 depicted in figure 5. One important issue within the context of dipole-dipole interactions (see section 5) is to be able to reduce the s-wave scattering length, here a 6, 6. Controlling the magnetic field at values close to the Feshbach resonance at 589 G with an accuracy on the order of ±2 mg (corresponding to a relative accuracy of δb/b ) would allow one to adjust a 6, 6 < a 0. 4 Bose-Einstein condensation of 52 Cr To achieve Bose-Einstein condensation of chromium [5], we choose slightly different powers in the two laser beams that form the crossed optical dipole trap. The horizontal laser beam has a maximum power of 9 W while the vertical one contains 4.75 W. Like in the Feshbach resonance measurements, a strategy of ramping down the power in the horizontal laser beam and ramp-

10 0 Stuhler, Griesmaier, Werner, Koch, Fattori, Pfau ing up the vertical laser beam power is used to further evaporatively cool the sample to degeneracy. The detailed ramps are described in [25] and will not be repeated here. Figure 6 shows the time dependence of the laser powers (left) and the route (phase-space density vs. atom loss) to Bose-Einstein condensation. The onset of Bose-Einstein condensation manifests itself in the Figure 6. The route to achieve Bose-Einstein condensation of chromium. Left: Time dependence of the laser powers for the final evaporation in the optical dipole trap. Right: Evolution of the phase-space density versus the number of remaining atoms. occurence of a two-component velocity distribution of the atomic cloud, which becomes visible in time-of-flight absorption images of the expanding condensate. Figure 7 shows three absorption images for different final laser powers Figure 7. Absorption images of chromium clouds taken 5 ms after release from the trap. From left to right, the final trap laser power is reduced which leads to lower temperatures and higher phase-space densities. (a) Thermal cloud at. µk, above the critical temperature T c 700 nk. (b) At T < T c, a peak in the density distribution signals the onset of Bose-Einstein condensation. (c) A nearly pure BEC of 50, 000 Cr atoms. that correspond to different final temperatures and phase-space densities. The critical temperature T c is on the order of 700 nk. We are able to produce nearly pure condensates with up to 00,000 chromium atoms.

11 Ultracold Cr: from Feshbach resonances to a dipolar BEC 5 Observation of magnetic dipole-dipole interaction in the Cr-BEC We start the investigation of the dipole-dipole interaction by preparing an almost pure Cr-BEC in which the atoms are fully polarized in the m s = 3 Zeeman substate. We then adiabatically change the laser intensities to form a nearly cigar shaped trap with experimentally determined frequencies of ω x /2π = 942(6) Hz, ω y /2π = 72(4) Hz and ω z /2π = 28(7) Hz oriented along the horizontal z-direction. The orientation of the atomic dipoles µ with respect to the trap axes is determined by a homogeneous external magnetic field B with absolute value B 2 G. Initially, B is parallel to the vertical y- direction and the atoms are maximally polarized along B ( µ B e y ). We either keep this orientation of µ or slowly rotate the magnetic field within 40 ms to the z-direction which also rotates the magnetic moments and results in µ e z. After an additional holding time of several milliseconds, we release the atom cloud from the trap by switching off the trap laser beams. We image the cloud after a variable time of free flight. Repeating the whole experimental sequence (Cr-BEC preparation, forming the anisotropic trap, eventually changing the orientation of the dipole moments, free flight of the condensate and imaging) many times, we are able to record the expansion dynamics for both orientations of the atomic dipole moments. Figure 8 shows these data sets. We y z 3 aspect ratio R y / Rz , B along y: B, B along z: B time of flight [ms] Figure 8. Aspect ratio of an expanding Cr-BEC for two different orientations of the atomic dipole moments µ defined by the direction of a homogeneous magnetic field B. Blue: experimental data (circles) and theoretical prediction without adjustable parameter (solid line) for µ e y. Red: data (diamonds) and theory curve (solid line) for µ e z ( µ parallel to the weak trap axis). Blue upward and red downward triangles are results of 3 measurements taken 0 ms after release for the corresponding orientations of µ. Dashed line: theory curve without dipole-dipole interaction. Inset: sketch of the in-trap BEC. Right axis: absorption images of the BEC for some aspect ratios. plot the aspect ratio defined as the extension of the BEC along the y-direction divided by its extension in z-direction. This is a convenient measurement quan-

12 2 Stuhler, Griesmaier, Werner, Koch, Fattori, Pfau tity since it depends only weakly on the atom number but is sensitive to the trap frequencies and the ratio between contact and dipole-dipole interaction. The data show a clear difference in the expansion of the Cr-BEC for the two orientations of µ, a manifestation of the dipole-dipole interaction [35]. The absolute values of the aspect-ratio are in excellent agreement with the theoretical prediction (straight lines in the figure) that is obtained without any free parameter and based on the theory of dipolar quantum gases. 6 Summary In this article, we have summarized three milestone experiments towards studies of dipole-dipole interaction in degenerate quantum gases. We have detected fourteen Feshbach resonances in collisions of ultracold Cr atoms in the magnetic field range of G. The eleven strongest ones build a complete set of resonances expected for dipole-dipole coupling up to second order. The average agreement between experimental and theoretical position of the resonances is better than 0.6 G. The latter are obtained by fitting the results of multichannel scattering calculations to the experimental data. The fitting routine comprises five adjustable parameters to model the molecular potentials (three scattering lengths a 6, a 4 and a 2 one for each involved molecular potential and two common dispersion coefficients C 6 and C 8 ) and results in best fit values for those parameters. Width and loss coefficient of two resonances are promising for Feshbach tuning of the contact interaction. This is of great importance for future experiments with chromium Bose-Einstein condensates. We have produced such Cr-BECs with up to 00,000 condensed atoms. Chromium atoms have a magnetic dipole moment that is large enough to cause experimentally visible dipole-dipole effects in the BEC. Indeed, we have observed such a dipole-dipole effect by studying the expansion of the Cr- BEC. We measure a modification of the condensates aspect ratio that depends on the orientation of the atomic magnetic dipole moments. This is the first manifestation of dipole-dipole interaction in a gas and can be interpreted as magnetostriction, a change in the shape of the trapped or expanding atomic cloud caused by a homogeneous external magnetic field. The results are consistent with the theory of dipolar quantum gases and prove that a Cr-BEC is an excellent model system to study dipole-dipole interaction in degenerate gases. Utilizing the Cr-BEC with its strong magnetic dipole-dipole interaction together with the Feshbach tuning technique to adjust the contact interaction and/or spinning mangetic fields to tune the dipole-dipole interaction [36], allows one to address many dipolar phenomena. For instance, one can probe the predicted stability diagram of dipolar BECs, which depends on the shape of

13 Ultracold Cr: from Feshbach resonances to a dipolar BEC 3 the trap and the direction of the dipole moments with respect to it, as well as on the ratio between contact and dipole-dipole interaction. Furthermore, the strong dipolar relaxation processes [29], which are connected to the large magnetic moment of chromium and prevented Bose-Einstein condensation of chromium in a magnetic trap, can be used to cool a sample by a continuous demagnetization cooling technique [37]. Due to its large spin S = 3, a Cr-BEC is also a very exciting system in the context of spinor BECs [38, 39]. Acknowledgements We thank A. Simoni, E. Tiesinga, P. Pedri, S. Giovanazzi and L. Santos for many fruitful discussions and comments. A. Simoni and E. Tiesinga have performed the assignment of the observed Feshbach resonances and the multichannel scattering calculations to extract the molecular potential parameters. The theory curves regarding the expansion of the dipolar Cr-BEC were calculated by P. Pedri, S. Giovanazzi and L. Santos. This work is funded by the German Science Foundation (DFG) (SPP6 and SFB/TR 2). References [] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, et al., Reviews of Modern Physics (999). [2] L. P. Pitaevskii and S. Stringari, Bose-Einstein condensation (Oxford University Press, 2003). [3] J. R. Anglin and W. Ketterle, Nature 46 2 (2002), and references therein. [4] E. Tiesinga, B. J. Verhaar and H. T. C. Stoof, Phys. Rev. A (993). [5] S. Jochim, M. Bartenstein, A. Altmeyer, et al., Science (2003). [6] M. W. Zwierlein, C. A. Stan, C. H. Schunck, et al., Phys. Rev. Lett (2003). [7] M. Greiner, C. A. Regal and D. Jin, Nature (2003). [8] K. E. Strecker, G. B. Partridge, R. I. Kamar, et al., in Atomic Physics 9, Proceedings of the Nineteenth International Conference on Atomic Physics, edited by L. G. Marcassa, K. Helmerson and V. S. Bagnato, pp (American Institute of Physics Conference Proceedings, vol. 770, New York, 2005, 2005). [9] M. W. Zwierlein, C. A. Stan, C. H. Schunck, et al., Phys. Rev. Lett (2004). [0] M. Bartenstein, A. Altmeyer, S. Riedl, et al., Phys. Rev. Lett (2004). [] M. Greiner, C. A. Regal and D. S. Jin, Phys. Rev. Lett (2005). [2] G. B. Partridge, K. E. Strecker, R. Kamar, et al., Phys. Rev. Lett (2005). [3] T. Bourdel, L. Khaykovich, J. Cubizolles, et al., Phys. Rev. Lett (2004). [4] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, et al., Nature (2005). [5] A. Griesmaier, J. Werner, S. Hensler, et al., Phys. Rev. Lett (2005). [6] M. Baranov, L. Dobrek, K. Góral, et al., Physica Scripta T02 74 (2002), and references therein. [7] J. Doyle, B. Friedrich, R. V. Krems, et al., Eur. Phys. J. D 3 49 (2004), and references therein as well as following articles in this special issue. [8] S. Giovanazzi, A. Görlitz and T. Pfau, J. Opt. B: Quantum Semiclass. Opt. 5 S208 (2003). [9] L. Santos, G. V. Shlyapnikov and M. Lewenstein, Phys. Rev. Lett (2003). [20] K. Góral, K. Rz ażewski and T. Pfau, Phys. Rev. A (R) (2000). [2] S. Yi and L. You, Phys. Rev. A (2000). [22] L. Santos, G. V. Shlyapnikov, P. Zoller, et al., Phys. Rev. Lett (2000). [23] C. Eberlein, S. Giovanazzi and D. H. J. O Dell, Phys. Rev. A (2005). [24] K. Góral, L. Santos and M. Lewenstein, Phys. Rev. Lett (2002). [25] A. Griesmaier, J. Stuhler and T. Pfau, Appl. Phys. B (2005), available online, DOI:0.007/s [26] J. Stuhler, P. O. Schmidt, S. Hensler, et al., Phys. Rev. A (R) (200).

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