⇦Chinese Physics Letters, 2019, Vol. 36, No. 7, Article code 073401 Measurement of S-Wave Scattering Length between $^6$Li and $^{88}$Sr Atoms Using Interspecies Thermalization in an Optical Dipole Trap * Xiao-Bin Ma (马晓彬)1, Zhu-Xiong Ye (叶祝雄)1, Li-Yang Xie (谢礼杨)1, Zhen Guo (郭臻)1, Li You (尤力)1,2**, Meng Khoon Tey (郑盟锟)1,2** Affiliations 1State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084 2Collaborative Innovation Center of Quantum Matter, Beijing 100084 Received 16 May 2019, online 20 June 2019; Erratum Chin. Phys. Lett. 36 (2019) 109902 *Supported by the National Key Research and Development Program of China under Grant Nos 2018YFA0306503 and 2018YFA0306504, and the National Natural Science Foundation of China under Grant Nos 91636213, 91736311, 11574177, 91836302 and 11654001.
**Corresponding author. Email: lyou@mail.tsinghua.edu.cn; mengkhoon_tey@mail.tsinghua.edu.cn Citation Text: Ma X B, Ye Z X, Xie L Y, Guo Z and You L et al 2019 Chin. Phys. Lett. 36 073401    Abstract We report the creation of the first mixture of $^6$Li and $^{88}$Sr atoms in an optical dipole trap. Using this mixture, a measurement of the interspecies thermalization process is carried out and the previously unknown interspecies s-wave scattering length between $^6$Li and $^{88}$Sr atoms is extracted to be $|a_{\rm ^6Li-^{88}Sr}|=(380^{+160}_{-100})a_0$ with $a_0$ being the Bohr radius from the rate of interspecies thermalization. DOI:10.1088/0256-307X/36/7/073401 PACS:34.50.Cx, 67.85.-d, 67.85.Pq © 2019 Chinese Physics Society Article Text Quantum gas mixtures composed of different chemical elements or isotopes of the same elements add many exciting possibilities to the study of various physics phenomena, e.g., heteronuclear Efimov resonance,[1,2] impurity in superfluids,[3–5] and ultracold chemical reactions.[6] Of particular interest, these mixtures constitute the starting point of forming heteronuclear ground-state molecules, which can possess large electric dipole moments and thus can exhibit long-range and anisotropic interactions,[7–9] facilitating rich applications ranging from precision measurements,[10,11] simulation of many-body quantum systems,[10,12] to quantum computation processing.[13] However, most quantum-gas-mixture experiments involve two alkali-metal species, research in this field has also extended to richer combinations[14–23] along with the development of cooling techniques for atoms beyond alkali elements.[24–30] Among them, mixtures of alkali and alkaline-earth(-like) atoms highlight a unique advantage for their possibilities of creating open-shell ground-state molecules with an electric dipole moment and an electron spin, which make them promising candidates for testing fundamental symmetries[31] and for simulating lattice-spin models.[32] To date, intensive efforts have been carried out in investigating mixtures of Li-Yb,[15,16] Rb-Sr,[17,18,33] Rb-Yb,[19,20] and Cs-Yb[21] atoms. The work we report here focuses on Li-Sr atom mixtures. Although a number of theoretical predictions on the properties of Li-Sr mixture have been reported[34–37] in recent years, few direct experimental measurements have been carried out. In this Letter, we report the first production of an optically trapped mixture of $^6$Li and $^{88}$Sr atoms based on a sequential cooling and trapping strategy for the two species. Using this system, we study thermalization between the two atomic species in the same optical dipole trap, and extract the previously unknown interspecies scattering length from the rate of thermalization. Given the low vapor pressure of Li and Sr atoms at room temperatures, we load both $^6$Li and $^{88}$Sr atoms into a magneto-optical trap (MOT) from decelerated atomic beams. Figure 1(a) illustrates the design of our vacuum setup, which is made up of a dual-species oven, a spin-flip Zeeman slower (ZS), and an octagonal glass science chamber. Typically, the Li and Sr ovens are heated to 450 $^{\circ}\!$C. The divergence angle of the atomic beam emitted from the oven is limited by an array of microtubes with an inner diameter of 200 µm and length of 10 mm. In our design, the two species share the same Zeeman slower, which features a capture velocity of 974 m/s (669 m/s) and deceleration coefficient of 0.43 (0.40) for $^6$Li ($^{88}$Sr) atoms.[38] The customized glass cell made by synthetic quartz plates with double-sided anti-reflection coating offers excellent optical access and allows for precise and fast magnetic-field control. To cool $^6$Li atoms, we adopt the standard MOT procedure using the $^2\!S_{1/2} \rightarrow ^2\!\!\!P_{3/2}$ (D2) transition at 671 nm with subsequent gray molasses (GM) cooling stage using the $^2\!S_{1/2} \rightarrow ^2\!\!\!\!P_{1/2}$ (D1) transition.[39,40] For $^{88}$Sr atoms, the presence of singlet and triplet states enables an intriguing two-stage cooling process, which starts with a blue MOT (bMOT) operating on the broad $^1\!S_{0} \rightarrow^1\!\!\!\!P_{1}$ transition at 461 nm (linewidth ${\it \Gamma}/2\pi=30.5$ MHz) to capture atoms from the slowed atomic beam, and ends with a red MOT (rMOT) on the narrow $^1\!S_{0} \rightarrow^3\!\!\!P_{1}$ transition at 689 nm (${\it \Gamma}/2\pi=7.5$ kHz) to cool atoms to µK-regime. During the blue MOT stage, we use a pair of repumping lasers at 679 nm and 707 nm wavelengths for recycling atoms falling into the $5s5p^3\!P_2$ dark states back into the MOT.
cpl-36-7-073401-fig1.png
Fig. 1. Li-Sr mixture setup. (a) Three-dimensional illustration of the main vacuum system, consisting of three major parts: a dual-species oven, a common spin-flip Zeeman slower, and an octagonal glass science chamber. (b) Top view of the optical layout near the science chamber. To provide better optical access, fiber couplers (not shown), dichroic mirrors (DM), and polarization beam splitters (PBS) are used extensively to combine various light beams. Some optical elements are omitted for clarity.
As the optimal magnetic field gradients for loading the Li and Sr atoms, and for performing the gray molasses and rMOT are very different, we employ a sequential cooling and trapping strategy to prepare a mixture of $^6$Li and $^{88}$Sr atoms in an optical dipole trap (ODT), which is formed by a horizontal 1064-nm light beam with a waist of 28 µm. Figure 1(b) depicts the top view of the optical layout near the science chamber, which shows various light beams needed for cooling and trapping the atoms. Figure 2 illustrates the simplified experimental sequence we adopt for preparing the $^6$Li-$^{88}$Sr mixture in the ODT. Our experiment starts with accumulating $^{88}$Sr atoms in the blue MOT for 500 ms, followed by a 250-ms rMOT cooling stage as described in detail in Ref.  [38]. By adjusting the magnetic bias field at the end of the rMOT stage, we optimize the overlap between the ODT and the rMOT, and transfer about $1.2\times10^7$ $^{88}$Sr atoms into the ODT with a typical trap depth of $k_{_{\rm B}}$ (82.4 µK) ($k_{_{\rm B}}$ being the Boltzmann constant). Then, we turn on the D2-line MOT for $^6$Li for a variable time. At the end of the D2-line MOT, a 1-ms D1-line GM is set up to further cool $^6$Li atoms to 38 µK. The $^6$Li atoms are optically pumped to the lowest hyperfine manifold $F=1/2$ at the end of the GM. As the GM calls for essentially zero external magnetic field, bias fields could no longer be used to adjust the position of the cooled $^6$Li atomic cloud. Therefore, the ODT is pre-adjusted to a position that provides optimal loading efficiency for $^6$Li atoms. To load $^6$Li atoms into the ODT, we adiabatically (for the Sr atoms in the trap) ramp up the ODT power to 26.4 W in 50 ms shortly before the gray molasses stage, raising the trap depth to $k_{_{\rm B}}$ (1.33 mK). At the end, we obtain optically trapped $^{6}$Li atoms with a temperature of 163 µK. At this point, a $^6$Li-$^{88}$Sr mixture in an ODT is created with $^6$Li atoms distributed equally in the two Zeeman sub-states $|F=1/2, M_F=\pm1/2\rangle$ and $^{88}$Sr atoms in the $^1\!S_0$ ground state. The number and temperature of atoms are measured via the standard absorption imaging technique using the $^2\!S_{1/2} \rightarrow^2\!\!\!P_{3/2}$ transition at 671 nm for Li and $^1\!S_{0} \rightarrow^1\!\!\!P_{1}$ transition at 461 nm for Sr.
cpl-36-7-073401-fig2.png
Fig. 2. The adopted experimental sequence for preparing $^6$Li-$^{88}$Sr mixtures and for measuring interspecies thermalization in an ODT. Here $^{88}$Sr and $^6$Li atoms are cooled and transferred to the ODT in sequence. The MOT magnetic field gradient, optical dipole trap laser power, and magnetic bias fields are varied over time to optimize for loading of atoms into the ODT. Interspecies thermalization is observed over a variable mixture holding time at a constant ODT depth.
cpl-36-7-073401-fig3.png
Fig. 3. Atom numbers of $^{88}$Sr (blue squares) and $^{6}$Li (red circles) in the ODT as a function of $^{6}$Li MOT loading time. For comparison, $^{88}$Sr atom number in the same MOT loading process but without $^{6}$Li atoms is also shown (open blue diamonds). The blue solid (dashed) line is an exponential-decay fit to $^{88}$Sr atom number, yielding a $1/e$ lifetime of 2.2 s (9.7 s) with (without) $^{6}$Li atoms in MOT. The error bars are smaller than the size of the symbols.
The $^{88}$Sr atoms in the ODT are observed to be influenced by the $^6$Li MOT. Figure 3 compares the loss of $^{88}$Sr atoms from the ODT as a function of the Li-MOT loading time with and without $^6$Li atoms. The latter is achieved by either switching off the cooling or the repumping light, or the magnetic gradient field, all end up with the same results. It is apparent that the existence of $^6$Li atoms in the MOT results in a dramatic reduction of the trapping lifetime for $^{88}$Sr atoms (2.2 s) compared to that without $^6$Li atoms (9.7 s). The latter is presumably due to collision with residual gases in the vacuum chamber and also to the plain evaporation of $^{88}$Sr atoms in the ODT. The dominant loss mechanism behind the former may be attributed to light assisted collisions between $^{88}$Sr atoms with electronically excited $^6$Li atoms.[41] During the process of increasing the ODT laser power for loading Li atoms, $^{88}$Sr atoms are heated from 23 µK to 98 µK, presumably due to adiabatic compression of the $^{88}$Sr atomic cloud. We control the ratio of the two species in the ODT, $N_{\rm Li}$/$N_{\rm Sr}$, by varying the loading time of the $^6$Li MOT.
cpl-36-7-073401-fig4.png
Fig. 4. Interspecies thermalization. Temperature evolutions of $^6$Li (red circles) and $^{88}$Sr (blue squares) atoms are shown as a function of holding time in the ODT. The red solid line shows the fitted result of a numerical model (see text). The inset shows the atom number of the two species as a function of the holding time. Both species experience negligible loss in 400 ms.
To give a first estimate of the scattering cross section between the two species, we study the interspecies thermalization by monitoring the evolution of the atom numbers and temperatures of both $^6$Li and $^{88}$Sr atoms after they become mixed in the ODT. The results are shown in Fig. 4. The initial atom number and temperature of $^6$Li ($^{88}$Sr) atoms are $N=8.3\times10^5$ ($8.4\times10^6$) and $T=163$ µK (98 µK), respectively, with an associated peak density of $1.7\times10^{10}$ cm$^{-3}$ ($2.8\times10^{11}$ cm$^{-3}$). The hotter $^6$Li atoms get cooled down and reach thermal equilibrium with the colder $^{88}$Sr atoms in approximately 250 ms, while $^{88}$Sr atoms are only slightly heated due to their much larger abundance. Note that the fermionic $^6$Li atoms in the $F=1/2$ hyperfine ground state experience essentially no intrastate scattering and a negligible interstate scattering at zero magnetic field.[42] Also, the $^{88}$Sr atoms exhibit a very small intrastate s-wave scattering length of $-2a_0$. We can, therefore, conclude that the intra-species and interspecies thermalization of the Li and Sr atoms in such a short duration must arise from the interspecies scattering between Li and Sr atoms. As for atom numbers, neither $^6$Li nor $^{88}$Sr atoms suffer from significant losses during the thermalization process, which indicates that inelastic collisions between the two species are negligible under our experimental conditions. We extract the $^6$Li-$^{88}$Sr s-wave scattering length from the thermalization process shown in Fig. 4, assuming that the elastic collisions in our measurements ($T_{\rm Li(Sr)} < $170 µK) are dominated by s-wave interactions. This assumption is reasonable given that the p-wave barrier threshold is $\sim $2.3 mK, according to the $C_6$ coefficient for LiSr.[43] The thermalization can be modeled by connecting the rate of interspecies thermalization to the interspecies s-wave scattering cross section $\sigma$ through the equation[44–47] $$ -\frac{1}{\Delta T}\frac{d(\Delta T)}{dt}=\frac{\xi}{\alpha}\bar{n}\sigma \bar{v},~~ \tag {1} $$ where $\Delta T=T_1-T_2$ is the temperature difference of the two species, $\alpha=2.7$ is the average times of collisions needed for thermalization between colliding pairs with equal mass,[48] and between nonequal mass partners it should be corrected by the factor $\xi=\frac{4 m_1 m_2}{(m_1+m_2)^2}$ ($\xi=0.24$ for $^{6}$Li-$^{88}$Sr mixture). The mean relative velocity is $\bar{v}=\sqrt{\frac{8k_{_{\rm B}}}{\pi}(\frac{T_1}{m_1}+\frac{T_2}{m_2})}$, and the overlap density is defined by $\bar{n}=(\frac{1}{N_1}+\frac{1}{N_2})\int n_1(\boldsymbol{r})n_2(\boldsymbol{r} )d^3r$ with $$ \int n_1(\boldsymbol{r})n_2(\boldsymbol{r})d^3r=\frac{N_1 N_2(m_1 m_2 \bar{\omega}^2_1 \bar{\omega}^2_2)^{3/2}}{[2\pi k_{_{\rm B}}(m_1\bar{\omega}^2_1 T_2+m_2\bar{\omega}^2_2 T_1)]^{3/2}},~~ \tag {2} $$ where $\bar{\omega}_{i}=(\omega_{xi}\omega_{yi}\omega_{zi})^{1/3}$ is the mean trap frequency for atom species $i$. To obtain the interspecies scattering cross section $\sigma$, we first perform an exponential fit to the temperature evolution of the $^{88}$Sr atoms. With this information, Eq. (1) can be solved numerically for the temperature evolution of the Li atoms at a given scattering cross section $\sigma$ since all other initial conditions are known. We obtain the interspecies cross section by varying $\sigma$ to give a minimum variance between the simulated Li temperatures and the measured results. The cross section $\sigma$ is related to the s-wave scattering length $a$ by the expression[49] $\sigma=4\pi a^2/(1+k^2 a^2)$ with $\hbar k$ being the relative collision momentum. Considering the lack of spin for $^{88}$Sr atoms, we expect $a_{\rm ^6Li-^{88}Sr}$ to be the same for the two states of $^6$Li atoms. Our best fitted results give an s-wave scattering length amplitude of $|a_{\rm ^6Li-^{88}Sr}|=(380^{+160}_{-100})a_0$. Under the model we employed, the main uncertainty in the measured scattering length comes from the systematic uncertainties in the trap frequencies at the level of about 10%. In conclusion, we have produced the first optically trapped mixture of $^6$Li and $^{88}$Sr atoms, and have investigated their interspecies thermalization experimentally. We extract the previously unknown scattering length between $^6$Li and $^{88}$Sr atoms to be $|a_{\rm ^6Li-^{88}Sr}|=(380^{+160}_{-100})a_0$ by numerically modeling the thermalization process. Our results show that efficient sympathetic cooling between Li and Sr atoms is possible and pave the way for possible double degenerate mixture of Li and Sr atoms. References Observation of Heteronuclear Atomic Efimov ResonancesLow-energy properties of three resonantly interacting particlesCollisional Stability of K 40 Immersed in a Strongly Interacting Fermi Gas of Li 6 Self-localization of a small number of Bose particles in a superfluid Fermi systemBound states of a localized magnetic impurity in a superfluid of paired ultracold fermionsMolecules near absolute zero and external field control of atomic and molecular dynamicsOptical Production of Ultracold Polar MoleculesFormation of Ultracold Polar Molecules in the Rovibrational Ground StateIntroduction to Ultracold Molecules: New Frontiers in Quantum and Chemical PhysicsPrecision Test of Mass-Ratio Variations with Lattice-Confined Ultracold MoleculesStrongly Correlated 2D Quantum Phases with Cold Polar Molecules: Controlling the Shape of the Interaction PotentialQuantum Computation with Trapped Polar MoleculesDouble-degenerate Bose-Fermi mixture of strontiumQuantum Degenerate Mixtures of Alkali and Alkaline-Earth-Like AtomsQuantum degenerate mixture of ytterbium and lithium atomsQuantum degenerate mixtures of strontium and rubidium atomsPhotoionization loss in simultaneous magneto-optical trapping of Rb and SrScattering lengths in isotopologues of the RbYb systemDegenerate Bose-Fermi mixtures of rubidium and ytterbiumProduction of a degenerate Fermi-Fermi mixture of dysprosium and potassium atomsDipolar Quantum Mixtures of Erbium and Dysprosium AtomsSpin-Singlet Bose-Einstein Condensation of Two-Electron AtomsBose-Einstein Condensation of ChromiumBose-Einstein Condensation of Alkaline Earth Atoms: Ca 40 Bose-Einstein Condensation of StrontiumBose-Einstein Condensation of Sr 84 Strongly Dipolar Bose-Einstein Condensate of DysprosiumBose-Einstein Condensation of ErbiumElectron electric-dipole-moment searches based on alkali-metal- or alkaline-earth-metal-bearing moleculesA toolbox for lattice-spin models with polar moleculesGround state of the polar alkali-metal-atom–strontium molecules: Potential energy curve and permanent dipole momentStudy of dipole moments of LiSr and KRb molecules by quantum Monte Carlo methodsAb initio study of ground and excited states of 6 Li 40 Ca and 6 Li 88 Sr moleculesTheoretical electronic structure of the molecules SrX (X = Li, Na, K) toward laser cooling studyA dual-species magneto-optical trap for lithium and strontium atoms Λ -enhanced sub-Doppler cooling of lithium atoms in D 1 gray molassesEfficient all-optical production of large Li 6 quantum gases using D 1 gray-molasses coolingLoading an optical dipole trapElastic and inelastic collisions of 6 Li atoms in magnetic and optical trapsElectric dipole polarizabilities at imaginary frequencies for hydrogen, the alkali–metal, alkaline–earth, and noble gas atomsObservation of a Zero-Energy Resonance in Cs-Cs CollisionsSympathetic Cooling with Two Atomic Species in an Optical TrapSympathetic Cooling in an Optically Trapped Mixture of Alkali and Spin-Singlet AtomsInterspecies thermalization in an ultracold mixture of Cs and Yb in an optical trapMeasurement of Cs-Cs elastic scattering at T =30 μKFeshbach resonances in ultracold gases
[1] Barontini G, Weber C, Rabatti F, Catani J, Thalhammer G, Inguscio M and Minardi F 2009 Phys. Rev. Lett. 103 043201
[2] Kuhnle E D, Ulmanis J, Häfner S and Weidemüller M 2016 Natl. Sci. Rev. 3 174
[3] Spiegelhalder F M, Trenkwalder A, Naik D, Hendl G, Schreck F and Grimm R 2009 Phys. Rev. Lett. 103 223203
[4] Targońska K and Sacha K 2010 Phys. Rev. A 82 033601
[5] Vernier E, Pekker D, Zwierlein M W and Demler E 2011 Phys. Rev. A 83 033619
[6] Krems R V 2005 Int. Rev. Phys. Chem. 24 99
[7] Sage J M, Sainis S, Bergeman T and DeMille D 2005 Phys. Rev. Lett. 94 203001
[8] Deiglmayr J, Grochola A, Repp M, Mörtlbauer K, Glück C, Lange J, Dulieu O, Wester R and Weidemüller M 2008 Phys. Rev. Lett. 101 133004
[9]Ni K K , Ospelkaus S, de Miranda M H G, Pe'er A, Neyenhuis B, Zirbel J J, Kotochigova S, Julienne P S, Jin D S and Ye J 2008 Science 322 5899 231
[10] Jin D S and Ye J 2012 Chem. Rev. 112 4801
[11] Zelevinsky T, Kotochigova S and Ye J 2008 Phys. Rev. Lett. 100 043201
[12] Büchler H P, Demler E, Lukin M, Micheli A, Prokof'ev N, Pupillo G and Zoller P 2007 Phys. Rev. Lett. 98 060404
[13] DeMille D 2002 Phys. Rev. Lett. 88 067901
[14] Tey M K, Stellmer S, Grimm R and Schreck F 2010 Phys. Rev. A 82 011608
[15] Hara H, Takasu Y, Yamaoka Y, Doyle J M and Takahashi Y 2011 Phys. Rev. Lett. 106 205304
[16] Hansen A H, Khramov A, Dowd W H, Jamison A O, Ivanov V V and Gupta S 2011 Phys. Rev. A 84 011606
[17] Pasquiou B, Bayerle A, Tzanova S M, Stellmer S, Szczepkowski J, Parigger M, Grimm R and Schreck F 2013 Phys. Rev. A 88 023601
[18] Aoki T, Yamanaka Y, Takeuchi M, Torii Y and Sakemi Y 2013 Phys. Rev. A 87 063426
[19] Borkowski M, Żuchowski P S, Ciuryło R, Julienne P S, Kȩdziera D, Mentel Ł, Tecmer P Münchow F, Bruni C and Görlitz A 2013 Phys. Rev. A 88 052708
[20] Vaidya V D, Tiamsuphat J, Rolston S L and Porto J V 2015 Phys. Rev. A 92 043604
[21]Kemp S L, Butler K L, Freytag R, Hopkins S A, Hinds E A, Tarbutt M R and Cornish S L 2016 Rev. Sci. Instrum. 87 2 023105
[22] Ravensbergen C, Corre V, Soave E, Kreyer M, Kirilov E and Grimm R 2018 Phys. Rev. A 98 063624
[23] Trautmann A, Ilzhöfer P, Durastante G, Politi C, Sohmen M, Mark M J and Ferlaino F 2018 Phys. Rev. Lett. 121 213601
[24] Takasu Y, Maki K, Komori K, Takano T, Honda K, Kumakura M, Yabuzaki T and Takahashi Y 2003 Phys. Rev. Lett. 91 040404
[25] Griesmaier A, Werner J, Hensler S, Stuhler J and Pfau T 2005 Phys. Rev. Lett. 94 160401
[26] Kraft S, Vogt F, Appel O, Riehle F and Sterr U 2009 Phys. Rev. Lett. 103 130401
[27] Stellmer S, Tey M K, Huang B, Grimm R and Schreck F 2009 Phys. Rev. Lett. 103 200401
[28] de Escobar Y N M, Mickelson P G, Yan M, DeSalvo B J, Nagel S B and Killian T C 2009 Phys. Rev. Lett. 103 200402
[29] Lu M, Burdick N Q, Youn S H and Lev B L 2011 Phys. Rev. Lett. 107 190401
[30] Aikawa K, Frisch A, Mark M, Baier S, Rietzler A, Grimm R and Ferlaino F 2012 Phys. Rev. Lett. 108 210401
[31] Meyer E R and Bohn J L 2009 Phys. Rev. A 80 042508
[32] Zoller P 2006 Nat. Phys. 2 341
[33]Barbé V, Ciamei A, Pasquiou B, Reichsöllner L, Schreck F, Z̈uchowski P S and Hutson J M 2018 Nat. Phys. 14 9 881
[34] Guérout R, Aymar M and Dulieu O 2010 Phys. Rev. A 82 042508
[35] Guo S, Bajdich M, Mitas L and Reynolds P J 2013 Mol. Phys. 111 1744
[36] Gopakumar G, Abe M, Hada M and Kajita M 2013 J. Chem. Phys. 138 194307
[37] Zeid I, Atallah T, Kontar S, Chmaisani W, El-Kork N and Korek M 2018 Comput. Theor. Chem. 1126 16
[38] Ma X, Ye Z, Xie L, Guo Z, You L and Tey M K 2019 arXiv:1904.05230 [cond-mat.quant-gas]
[39] Grier A T, Ferrier-Barbut I, Rem B S, Delehaye M, Khaykovich L, Chevy F and Salomon C 2013 Phys. Rev. A 87 063411
[40] Burchianti A, Valtolina G, Seman J A, Pace E, De Pas M, Inguscio M, Zaccanti M and Roati G 2014 Phys. Rev. A 90 043408
[41] Kuppens S J M, Corwin K L, Miller K W, Chupp T E and Wieman C E 2000 Phys. Rev. A 62 013406
[42] Houbiers M, Stoof H T C, McAlexander W I and Hulet R G 1998 Phys. Rev. A 57 R1497
[43] Derevianko A, Porsev S G and Babb J F 2010 At. Data Nucl. Data Tables 96 323
[44] Arndt M, Ben Dahan M, Guéry-Odelin D, Reynolds M W and Dalibard J 1997 Phys. Rev. Lett. 79 625
[45] Mudrich M, Kraft S, Singer K, Grimm R, Mosk A and Weidemüller M 2002 Phys. Rev. Lett. 88 253001
[46] Ivanov V V, Khramov A, Hansen A H, Dowd W H, Münchow F, Jamison A O and Gupta S 2011 Phys. Rev. Lett. 106 153201
[47] Guttridge A, Hopkins S A, Kemp S L, Frye M D, Hutson J M and Cornish S L 2017 Phys. Rev. A 96 012704
[48] Monroe C R, Cornell E A, Sackett C A, Myatt C J and Wieman C E 1993 Phys. Rev. Lett. 70 414
[49] Chin C, Grimm R, Julienne P and Tiesinga E 2010 Rev. Mod. Phys. 82 1225