Feshbach spectroscopy of an ultracold ⁴¹K-⁶Li mixture and ⁴¹K atoms

Magnetic Feshbach resonance Chin et al. (2010) provides an essential tool in the study of ultracold atom physics. Several breakthroughs have been demonstrated by tuning the interaction strength with Feshbach resonance, such as the formation of solitons Khaykovich et al. (2002); Strecker et al. (2002), the observation of Efimov trimer states Kraemer et al. (2006); Zaccanti et al. (2009); Knoop et al. (2009), and the creation of polar molecules Ni et al. (2008). Of particular interests are the broad Feshbach resonances, which are useful for studying universal properties in few-body and many-body atomic systems. Broad s-wave Feshbach resonance offers a great opportunity to study both weak-coupling Bardeen-Cooper-Schrieffer (BCS) superfluid and Bose-Einstein condensation (BEC) of tightly bound fermion pairs Regal et al. (2004); Bourdel et al. (2004); Chin et al. (2004); Zwierlein et al. (2005). Broad high partial wave resonances hold great promise for the study of three-body recombination decay Suno et al. (2003), the formation of p-wave molecules Gaebler et al. (2007), and the realization of d-wave molecular superfluid Yao et al. (2017).

Up to now, Feshbach resonances have been studied in most of the laser-cooled atomic species Chin et al. (2010); Frisch et al. (2014); Maier et al. (2015). Among them, lithium and potassium atoms have attracted intense interests both in experimental and theoretical studies. Many of s- or p- wave resonances have been reported in the isotopes of lithium and potassium Jochim et al. (2002); Khaykovich et al. (2002); D’Errico et al. (2007); Loftus et al. (2002); Chen et al. (2016), especially two broad s-wave resonances in ⁶Li Jochim et al. (2003) and ⁴⁰K Greiner et al. (2003), respectively. Interspecies Feshbach resonances have been studied in the Fermi-Fermi mixture of ⁶Li and ⁴⁰K Wille et al. (2008); Tiecke et al. (2010). The potential functions of singlet and triplet states of lithium-potassium mixture have also been observed Tiemann et al. (2009). For the Bose-Fermi mixture of ⁴¹K-⁶Li, although superfluid mixture have been realized with the observation of vortex lattices Yao et al. (2016), a full description of its Feshbach resonance is still missing. Moreover, the recent observation of a broad d-wave shape resonance in the lowest energy channel of ⁴¹K atoms Yao et al. (2017) brings great promise for the investigation of universal physics with strong coupling in nonzero partial waves. Therefore, a more complete search on the d-wave resonances in ⁴¹K systems is urgently required.

In this work, we report on the first extensive experimental and theoretical study on the s- and p- wave Feshbach resonances in ⁴¹K-⁶Li mixture and d-wave Feshbach resonances in single species ⁴¹K. We observe 69 interspecies Feshbach resonances in 10 spin combinations of ⁴¹K-⁶Li mixture and 6 new broad d-wave resonances in 4 spin combinations of ⁴¹K atoms (see Fig. 1 for the interested spin channels). We perform semi-analytic multi-channel quantum defect theory (MQDT) calculation based on parameters reported in previous literatures Tiemann et al. (2009) to identify these resonances. After fine tuning of the singlet and triplet quantum defect parameters, perfect accordance are achieved between the theoretical and experimental results. In the case of interspecies Feshbach resonances, all the observed s-wave resonances are identified as narrow resonances, while several p-wave resonances possess quite wide resonance widths, which are also experimentally ascertained by the distinct doublet structure on the loss spectroscopy. For the single species ⁴¹K system, all the d-wave broad resonances possess fully resolved triplet structure on the loss spectroscopy, which exactly demonstrates that they are from real d-wave to d-wave coupling Yao et al. (2017); Cui et al. (2017). Our vast amounts of experimental data and improved scattering parameters are important to the future research on ⁴¹K-⁶Li mixture and ⁴¹K system.

This paper will be organized as following: In section II, we briefly introduce the experiment method for preparing the ultracold ⁴¹K and ⁶Li atoms and detecting the scattering resonances therein; In section III, the details of MQDT calculation are given; In section IV, we present the results of s-wave and p-wave resonances in ⁴¹K-⁶Li mixture, and d-wave resonances in single ⁴¹K system, respectively; In section V, a conclusion and outlook are drawn.

The experimental procedure to produce the ultracold ⁴¹K-⁶Li mixture has been described in our previous works Chen et al. (2016); Yao et al. (2016); Wu et al. (2017). After radio frequency (rf) evaporative cooling in the optically-plugged magnetic trap, we load about $9\times 10^{6}$ ⁶Li and $3\times 10^{6}$ ⁴¹K atoms into a cigar-shaped optical dipole trap and immediately prepare them in the lowest spin states, respectively. Then, we increase the magnetic field to 435 G and implement a 8 s forced evaporative cooling on ⁴¹K atoms by lowering the laser intensity, while the ⁶Li atoms are sympathetically cooled. Next, the cold mixture is adiabatically transferred into a disk-like optical dipole trap, where two elliptical laser beams with aspect ratio of 4:1 are crossed perpendicularly (wavelength 1064 nm, maximum laser power of each beam 1.1 W, $1/e^{2}$ axial (radial) radius 48 $\mu$m (200 $\mu$m)). Further evaporative cooling of 8 s is performed within the disk-like trap, where ultracold ⁴¹K-⁶Li mixture with targeted atom numbers and temperatures can be prepared by choosing appropriate final laser intensity. Finally, absorption imaging along the gravity direction is applied to probe the two species in a single experimental run.

For the s-wave Feshbach resonance measurement, we prepare the mixture at lowest temperature with $2.2\times 10^{5}$ ⁴¹K atoms at 60 nK or 0.65 BEC temperature ($T_{c}$) and $5.5\times 10^{5}$ ⁶Li atoms at 270 nK or 0.5 Fermi temperature ($T_{F}$), respectively, to reduce the temperature induced broaden and resonance position shift of the loss spectrum. We mention that the ⁶Li and ⁴¹K atoms cannot reach fully thermal equilibrium because the single spin ⁶Li atoms can only thermalize through collisions with ⁴¹K atoms and the cloud size difference between them is huge. Fortunately, we find this effect does not affect the Feshbach spectroscopy. For the p-wave resonance measurements, to allow atoms tunnelling through the centrifugal barrier, we prepare the mixture at higher temperatures with $1.4\times 10^{6}$ ⁴¹K atoms at 350 nK and $1\times 10^{6}$ ⁶Li atoms at 480 nK, respectively. Next, to prepare different spin state combinations, several Landau-Zener pulses are successively applied to transfer the ⁶Li $|a\rangle$ and ⁴¹K $|a\rangle$ atoms to the desired spin states with efficiencies higher than 99%.

To facilitate the measurements, we use the same experimental sequence to produce the ultracold ⁴¹K atoms, but without loading ⁶Li atoms at the magneto-optical trap stage. In order to detect the triplet splitting of d-wave resonances, the cloud is prepared at 310 nK, which is slightly below the $T_{c}$ Yao et al. (2017) with more than $1.8\times 10^{6}$ ⁴¹K atoms. Then, we fine tune the rf power of Landau-Zener pulses to prepare the targeted spin combinations of ⁴¹K atoms. We mention that the background lifetime for all the spin combinations of ⁴¹K-⁶Li mixture and pure ⁴¹K are more than 10 s, even for the inelastic scattering channels, long enough for the following measurements.

After initial state preparation, inelastic loss spectroscopy is performed to detect the scattering resonances where enhanced atom losses occur due to three-body decay. For each measurement, to reduce the systematic errors, the magnetic field is ramped from 435 G to a few Gauss below or above the resonance in 50 ms, depending on the resonance position is below or above 435 G, respectively. Then, after a 100 ms holding time at that magnetic filed for equilibrium, the magnetic field is quickly switched to a desired value and hold for some time between 100 ms to 5 s. The waiting time is carefully chosen for different scattering resonances such that obvious atom loss signal can be acquired and detectable atoms still survive in the trap to reduce the saturation effects. Finally, the magnetic field is ramped back to 435 G to simultaneously detect the remaining ⁶Li and ⁴¹K atom number using resonant absorption imaging. It’s found that the loss signals of ⁶Li atoms are not as good as ⁴¹K atoms, mainly due to the large difference of cloud sizes between ⁶Li and ⁴¹K cloud. Therefore, we only adopt the results of ⁴¹K atoms to derive the parameters of scattering resonances. Moreover, additional measurements on single-species ⁶Li and ⁴¹K atoms are respectively taken to ensure the observed loss spectrums are solely contributed by the interspecies interactions.

To predict the position and other scattering parameters of the Feshbach Resonances, we adopt the semi-analytical MQDT theory developed in Gao (1998); Gao et al. (2005); Gao (2008). In this approach, two quantum defect functions $K^{c}_{S}(\epsilon,\ell),~{}K^{c}_{T}(\epsilon,\ell)$ are required as inputs where $\epsilon,~{}\ell$ are scattering energy and angular momentum, respectively. When completely neglecting the dependence on $\epsilon,~{}\ell$ and setting $K^{c}_{S}(\epsilon,\ell)=K^{c}_{S}(0,0),~{}K^{c}_{T}(\epsilon,\ell)=K^{c}_{T}(0,0)$, one can already obtain most of the resonance positions with reasonable agreement comparing to the experimental results Cui et al. (2017); Makrides and Gao (2014). However, as pointed out in Makrides and Gao (2014), including the $\ell$ dependence with a linear form in $K^{c}_{S}(\epsilon,\ell),~{}K^{c}_{T}(\epsilon,\ell)$ greatly improves the accuracy and extends the predicting power of this MQDT.

Considering the correction from the $\epsilon$ dependence is usually much smaller comparing to that from $\ell$ dependence, we choose the following form in the study of ⁴¹K-⁶Li resonances:

With such choice of parametrization for $K^{c}_{S}(\epsilon,\ell),~{}K^{c}_{T}(\epsilon,\ell)$, all scattering properties in all partial waves and all different hyperfine levels, can be determined by four parameters $K^{c}_{S}(0,0),~{}K^{c}_{T}(0,0)$ and $\beta_{S},~{}\beta_{T}$. In turn, these four parameters can be fixed with four different known resonance positions in two different $\ell$ (two resonances for each $\ell$) from experiments. In practice, we choose two s-wave resonance at $B=352.46~{}G$ (channel $|ba\rangle$) and $B=410.54~{}G$ (channel $|bc\rangle$) and two p-wave resonance at $B=210.09~{}G$ (channel $|ca\rangle$) and $B=239.71~{}G$ (channel $|cb\rangle$) to fix these four parameters. As a starting point, we first relate the parameters $K^{c}_{S,T}(0,0)$ to the singlet and triplet scattering length $a_{S},~{}a_{T}$ through the analytic formula Gao et al. (2005); Hanna et al. (2009):

If we take $a_{S}=42.75a_{0},~{}a_{T}=60.77a_{0}$ from early literature Tiemann et al. (2009), we get $K^{c}_{S}(0,0)=-3.2704,~{}K^{c}_{T}(0,0)=8.5430$. Taking these values of $K^{c}_{S,T}(0,0)$ to calculate the s-wave scattering, the predicted resonance positions will deviate from the experimental values from several Gauss to dozens of Gauss. Fine tunings of $K^{c}_{S,T}(0,0)$ are then necessary to get a better fit. Finally, we get $K^{c}_{S}(0,0)=-3.1235$ and $K^{c}_{T}(0,0)=8.9785$, which results in the new values $a_{S}=42.24a_{0}$, $a_{T}=60.48a_{0}$. Under these choices, the predicted s-wave resonance positions agree perfectly with experimental results, as shown in Table 1.

For the ⁴¹K-⁶Li p-wave resonances, the predicted resonance positions with fine tuned $K^{c}_{S,T}(0,0)$ still deviate from the experimental values res on the level of ten Gauss, if we simply neglect the $\ell$ dependence in $K^{c}_{S}(\epsilon,\ell)$ and set $\beta_{S}=\beta_{T}=0$ in Eq. (1) and (2). Then by further adjusting the values of $\beta_{S,T}$ to $~{}\beta_{S}=0.0455$, $\beta_{T}=0.3407$, we can also get perfect agreement for the p-wave resonances between theory and experimental results, as shown in Table 2. As for the ${}^{41}K$ resonances, since we focus on the d-wave resonances in this work, we only need two parameters $K^{c}_{S}(0,\ell=2)$ and $K^{c}_{T}(0,\ell=2)$ which is fixed by two d-wave resonance positions. By choosing the two resonances in the channel $|aa\rangle$ (see Table 3), we find $K^{c}_{S}(0,\ell=2)=2.0985,~{}K^{c}_{T}(0,\ell=2)=11.2949$. Similarly, one can also see the excellent quantitative agreement with the experimental data shown in Table 3.

Finally, we want to emphasize that, our MQDT approach is supposed to work much better on light alkali atoms while not so good for the heavy ones. The reason for this is two fold: a) light atoms usually have smaller hyperfine splitting energy comparing to heavy ones, which justifies the neglecting of energy dependence in $K^{c}_{S,T}(\epsilon,\ell)$; (b) the effects of anisotropic interactions including magnetic dipole-dipole interaction and second order spin-orbit coupling are more important in heavy atoms which are not taken into account in our MQDT calculations. As a result, we expect our MQDT to have a similar predicting power on resonances of light atoms like Li, Na, K and their mixtures while it may not be so accurate on heavy ones like Rb and Cs.

Besides the resonance position, several other parameters are required for a complete description of a well isolated resonance. Close to each $\ell$-wave resonance, we have:

where $a_{\ell}$ is the generalized $\ell$-wave “scattering length” or the so called scattering hyper-volume and $B_{res}$ is the position of resonance. Another two parameters $\zeta_{res}$ and $\delta\mu$ are defined through the leading order energy and magnetic field dependence of the quantum defect $K_{\ell}^{c0}(\epsilon,B)$ around $\epsilon=0$ and $B=B_{res}$ Gao (2011):

where $\delta B=B-B_{res}$, $w_{\ell}=[(2\ell+3)(2\ell-1)]^{-1},$ and $E_{6}=\hbar^{2}/(m\beta_{6}^{2})=E_{vdW}/4$.

The quantity $\zeta_{res}$ characterize whether the resonance is open-channel ($|\zeta_{res}|\gg 1$) or closed-channel ($|\zeta_{res}|\ll 1$) dominated. It was shown that $\zeta_{res}>0$ for s-wave and $\zeta_{res}<0$ for all high partial waves Gao (2011). For a Feshbach resonance, $\delta\mu$ defined in Eq. 5 approximately equals to the magnetic moment difference between the free atom pair and closed channel molecule. However, for a shape resonance, since there is no closed channel molecule, the value of $\delta\mu$ does not have such physical meaning. One can also show that $\zeta_{res},~{}a_{bg\ell},~{}\delta\mu$ and $\Delta_{B}$ are related as:

The above resonance parameters provide a complete description for each isolated resonance and are shown in the Table 1-3.

In conclusion, we have taken an intensive experimental study on the Feshbach resonances of ⁴¹K-⁶Li mixture and d-wave Feshbach resonances of ⁴¹K with the aid of MQDT calculation, and confirmed the resonance positions with very high accuracy. Several pretty wide p-wave resonances in ⁴¹K-⁶Li mixture and extremely broad d-wave resonances in single ⁴¹K system are discovered. Both p- and d- wave resonances have experimental resolvable doublet or triplet structure, respectively. The long background lifetime of the states possessing p- or d- wave resonances enable our system being an ideal platform to pursue many important non-zero partial wave related physics.