All-optical production of a superfluid Bose-Fermi mixture of Li and Li
We report the first all-optical production of a superfluid Bose-Fermi mixture with two spin states of Li (fermion) and one spin state of Li (boson) under the resonant magnetic field of the -wave Feshbach resonance of the fermions. Fermions are cooled efficiently by evaporative cooling and they serve as coolant for bosons. As a result, a superfluid mixture can be achieved by using a simple experimental apparatus and procedures, as in the case of the all-optical production of a single Bose-Einstein condensate (BEC). We show that the all-optical method enables us to realize variety of ultracold Bose-Fermi mixtures.
Quantum many-body systems of strongly interacting fermions are ubiquitous in condensed matter physics, nuclear physics, astrophysics, and atomic physics. Tunable interactions between particles through the use of a Feshbach resonance have enabled us to simulate such Fermi systems in cold atom experiments. To date, many-body physics for two-component fermions have been studied in the absence of impurities [1, 2]. However, realistic physical systems are not such a pure Fermi system. For example, phonons, holes, and impurities give rise to unique features in condensed matter at low temperature.
A Bose-Fermi mixture is an example of a system that allows us to investigate fermions interacting with other particles. Historically, Bose-Fermi mixtures have been studied for the purpose of cooling fermions via sympathetic cooling [3, 4]. However, recent breakthroughs in realizing superfluid Bose-Fermi mixtures of Li and Li  have completely changed the objective of studying such mixtures. Superfluid Bose-Fermi mixtures have the potential to reveal novel quantum phenomena such as the phase diagram and stability of structures [6, 7, 8, 9], changes in the dispersion relation of Bose-Einstein condensates (BECs) , the lifetime of quasiparticles, the damping rates of elementary excitations [10, 11, 12], counterflows of two superfluids and their critical velocities , and quantum phase transitions in Bose-Fermi mixtures [13, 14].
In this paper, we report the first all-optical production of a superfluid Bose-Fermi mixture of Li and Li. Previously, all-optical methods have been developed to realize a BEC  or a degenerate Fermi gas efficiently using a simple experimental apparatus . These methods are also compatible with the use of Feshbach resonance. However, until now, a superfluid Bose-Fermi mixture had not yet been obtained using all-optical methods.
2 Experimental setup
A part of the experimental setup is shown in Fig. 1. Li and Li are pre-cooled above a lithium oven using a two-dimensional MOT (2D-MOT), as shown in Ref. . They are loaded into a simultaneous three-dimensional magneto-optical trap (3D-MOT)  in a glass cell through a differential pumping tube. Cooling and repumping lasers for the simultaneous MOT have radii of 5 mm, and their laser power are mW, mW, mW, and mW. As suggested in Ref. , we use the D1 line only for Li repumping and the D2 lines for other transitions in order not to heat Li. The quadrupole magnetic field for the MOT is set to 18 Gauss/cm in the strong direction (the direction in the figure).
The optical setup for the optical dipole trap (ODT) is illustrated in Fig. 1. We use a double ODT system that can produce a deep ODT and a stable ODT . The light source of this double ODT system is a multimode ytterbium fiber laser with a maximum output power of 200 W, a center wavelength of 1070 nm, and a line width of 3 nm. The output is divided into two optical paths. One is used for the deep ODT produced with high laser power (HP-ODT). The power is controlled using an acousto-optic modulator (AOM) with a large active aperture of 2.52.5 mm manufactured in crystal quartz with a small absorptance of 0.4% (HP-AOM). The other path is used for the stable ODT produced by low laser power (LP-ODT). The power is controlled using an AOM with an active aperture of 11 mm manufactured in TeO (LP-AOM). The diffracted laser is coupled into an optical fiber for spatial filtering and pointing stability. Both laser beams are focused to the MOT region in the glass cell with beam waists of = 37 m and = 45 m. They are orthogonally polarized relative to each other to avoid interference at their intersection. The finite incident angles of = 16 and = 5 are chosen in order to prevent back reflections inside the glass cell from returning to the trapping region. The glass cell is made from anhydrous quartz to reduce thermal lens effects caused by the high-power laser ODT. The laser power of the LP-ODT is stabilized by monitoring the power after the glass cell and feeding the error signal back to the LP-AOM. Typically, a laser power of 100 W (2 W) can be delivered inside the glass cell used for the HP-ODT (LP-ODT) under an output of 160 W from the fiber laser. The maximum depth of these ODTs are 2.8 mK and 38 K, respectively. The measured 1/e lifetime of the LP-ODT is 60 s.
A bias magnetic field for the Feshbach resonance of Li is produced by the same pair of coils used for the MOT. They produce a magnetic curvature in the direction, corresponding to 0.24 Hz for Li and 0.24 Hz for Li, where is the bias magnetic field produced by the coils in units of Gauss. =0.93 is a factor for the mass difference. We omit magnetic curvature in the and directions since atomic confinement by the ODTs is dominant in these two directions. The total trapping potential is given by a combination of the ODTs and this magnetic confinement.
The probe laser beams for Li and Li are aligned on the path of the LP-ODT and applied along the direction for the measurement of the momentum distributions. Additionally, we apply probe beams along the direction to measure the in-situ density distributions of atoms .
3 Experimental procedure
The experimental procedure is shown in Fig. 2. We collected 210 Li atoms and 210 Li in the simultaneous MOT within 30 s. We loaded them into the HP-ODT by further cooling and compressing in a compressed-MOT (CMOT) for 45 ms. During the CMOT process, we gradually increased the MOT field up to 90 Gauss/cm, decreased the power of the MOT lasers, and changed the detuning of the MOT lasers in the presence of an HP-ODT with a depth of 2.8 mK. Approximately 110 Li atoms and 310 Li atoms are loaded into the HP-ODT in this loading process. This difference in loading efficiency between Li and Li is caused by a small position displacement of Li MOT from Li MOT.
Immediately after the CMOT, a bias magnetic field of 832.18 Gauss is turned on for subsequent evaporative cooling. At the beginning of evaporative cooling, Li atoms are populated in the two magnetic sublevels of the lowest hyperfine state , and Li atoms are populated in the three magnetic sublevels of the state. At this magnetic field, fermions have a diverging scattering length of for collisions between the two states , and bosons have for [22, 23], for [5, 24]. The scattering length of for is unknown. In this experiment, we do not consider the scattering length between different internal states of the bosons, because we choose one of these internal states in the process of evaporative cooling. The scattering lengths between bosons and fermions have almost the same values of for all the combinations of internal states [3, 5]. Since , , fermions can be cooled more efficiently than bosons by evaporative cooling. Thus, bosons can be cooled by thermal contact with cooled fermions, namely, by sympathetic cooling. This process is the reverse of the sympathetic cooling of fermions by bosons [3, 4].
Evaporative cooling was performed in the double ODT system. First, the trap depth of the HP-ODT was lowered from 2.8 mK to 560 K for a period of by decreasing the output laser power of the fiber laser from 160 W to 33 W, which changes the power inside the glass cell from 100 W to 20 W. We then applied a radio-frequency magnetic pulse for 300 ms to Li with a resonant frequency between the two magnetic sublevels of in order to populate them equally and maximize the efficiency of evaporative cooling. The state of Li disappeared during evaporative cooling owing to inelastic collisions among the bosons at this magnetic field, leaving the stable internal states and . We prepared one of these internal states by applying a blower laser pulse to Li atoms for 50 s with a frequency tuned to or to remove them selectively from the trap. After the state preparation process, we turned on the LP-ODT with a depth of 38 K (2 W) and produced a crossed-ODT at the intersection of the HP-ODT and LP-ODT and gradually turned off the HP-ODT for a period of . During this step, atoms were cooled and transferred into the stable LP-ODT. They were finally cooled by lowering the trap depth down to 850 nK (45 mW) for a period of and held for a period of until they reached thermal equilibrium. The final trap has trapping frequencies of Hz and Hz for Li in the radial and axial directions, respectively. The frequencies for Li were obtained by multiplying those for Li by a factor of .
The final state of the Bose-Fermi mixture was examined by measuring the momentum distributions of the paired fermions and the bosons. We turned off the LP-ODT and the bias magnetic field simultaneously (within 10 s of each other). After the Li atoms expanded ballistically during 8 ms of the time of flight (TOF), the probe laser beam was applied for absorption imaging. In contrast, the paired fermions of Li atoms were converted into molecules when the magnetic field was turned off [25, 26], and those molecules expanded with the original center-of-mass momentum distribution of the paired fermions. Although unpaired fermions also expanded as atoms or as randomly associated molecules, they exhibited a thermal distribution. Therefore, macroscopic occupation at the zero momentum state in the molecules suggested the existence of condensate paired fermions in the Fermi system before the magnetic field was turned off . Instead of imaging the molecules directly, they were dissociated into atoms by applying a magnetic pulse with a larger peak than the resonant magnetic field strength of the Feshbach resonance. In this way, the momentum distribution of the paired fermions and unpaired atoms were imaged after the total TOF time of 11 ms.
4 Experimental results
First, we show the experimental results of evaporative cooling for a mixture of fermions and bosons in the state with . To maximize the final phase space density of Li, we optimized the set , , , and . They were found to be 5 s, 4 s, 20 s, and 1 s, respectively. Fig. 3a shows the final number of atoms after evaporative cooling as a function of the initial mixing ratio given by , where and are the initial numbers of Li and Li at the beginning of evaporative cooling, respectively. Here, was fixed to , and only was controlled by the power of the Li cooling laser in the MOT. Fig. 3b shows the condensate fraction (CF) of the paired fermions and bosons, evaluated by fitting a bimodal distribution function to their momentum distributions (Fig. 3c). When the Li atoms were not mixed (), approximately Li atoms reached the superfluid state and the observed CF was approximately 0.6. This value is close to the maximum CF of the paired fermions at the unitarity limit . We can see that bosons can enter the BEC phase at . The fact that both Li and Li show finite CF at provides direct evidence for the presence of a superfluid Bose-Fermi mixture. At , the superfluid mixture consisted of Li and Li.
Next, we show the experimental results of evaporative cooling for a mixture of fermions and bosons in the state with an unknown scattering length . We investigated the final state by changing with fixed parameters of s, s, s, and . The result is shown in Fig. 4. When we chose s, fermions exhibited a bimodal distribution but bosons exhibited a thermal distribution (Fig. 4a,d,g). At s, which is the optimum condition for , boson and fermions both exhibited bimodal distributions in their momentum distributions (Fig. 4b,e). This is evidence for a superfluid Bose-Fermi mixture. However, the in-situ density distribution of bosons showed fragmented BECs (Fig. 4h), while the case of showed a single BEC (Fig. 4j). When we prolonged up to 40 s, the bosons in showed a single BEC (Fig. 4i). We can see that the axial size of the BEC is much smaller than that of the repulsively interacting BEC of (). We evaluated the scattering length to be at Gauss from the number of atoms in the BEC: , the axial size in the direction: m, and the trapping frequencies according to . The negative sign of the scattering length is consistent with the inequality of m, where is the harmonic oscillator length. Therefore, this is the first realization of a mixture of Fermi superfluid and an attractive BEC. The observed fragmented BECs can be explained as the spontaneous creation of Kibble-Zurek solitons, which was studied in Ref. .
When we prepared only Li without Li at the beginning of evaporative cooling, we could not increase the phase space density of Li even for a longer evaporation time. This means that the elastic scattering rate of Li determined by the initial phase space density and the scattering length is not sufficient for evaporative cooling. Thus, Li atoms in the Bose-Fermi mixture are mainly cooled by sympathetic cooling with fermions. Fig. 3a also shows evidence for sympathetic cooling. The figure shows that decreases as increases. This could be because more Li atoms need to be evaporated to cool more Li atoms by sympathetic cooling.
We demonstrated an experimental technique to produce a superfluid Bose-Fermi mixture of Li and Li using an all-optical method. During evaporative cooling, fermions were cooled efficiently by evaporative cooling with an enhanced scattering rate, while bosons were successfully cooled by sympathetic cooling with the aid of fermions. As a result, such a fascinating mixture can be produced using a simple experimental apparatus, as in the case of all-optical production of a single BEC. We demonstrated various Bose-Fermi mixtures by controlling the number, spin state, and cooling speed of bosons. Such high controllability paves the way for future studies on many-body physics of superfluid fermions with impurities.
The authors would like to thank J. Metz, K. Togashi, and T. Otsu for supporting the initial stage of this work. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas (Grant No. 24105006) and a Grant-in-Aid for Young Scientists (A) (Grant No. 23684033).
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