Magnetic Feshbach resonances allow control of the interactions between ultracold atoms1. They are an invaluable tool in studies of few-body and many-body physics2,3, and can be used to convert pairs of atoms into molecules4,5 by ramping an applied magnetic field across a resonance. Molecules formed from pairs of alkali atoms have been transferred to low-lying states, producing dipolar quantum gases6. There is great interest in making molecules formed from an alkali atom and a closed-shell atom such as ground-state Sr or Yb. Such molecules have both a strong electric dipole and an electron spin; they will open up new possibilities for designing quantum many-body systems7,8, and for tests of fundamental symmetries9. The crucial first step is to observe Feshbach resonances in the corresponding atomic mixtures. Very narrow resonances have been predicted theoretically10,11,12, but until now have eluded observation. Here we present the observation of magnetic Feshbach resonances of this type, for an alkali atom, Rb, interacting with ground-state Sr.
A magnetic Feshbach resonance arises when a pair of ultracold atoms couples to a near-threshold molecular state that is tuned to be close in energy by an applied magnetic field. Magnetoassociation at such a resonance coherently transfers the atoms into the molecular state13,14. In a few cases, near-threshold molecules formed in this way have been transferred to their absolute ground states15,16,17, allowing exploration of quantum gases with strong dipolar interactions6. However, this has so far been achieved only for molecules formed from pairs of alkali atoms.
Mixtures of closed-shell alkaline-earth atoms with open-shell alkali atoms have been studied in several laboratories18,19,20,21,22. No strong coupling mechanism between atomic and molecular states exists in systems of this type, but theoretical work has identified weak coupling mechanisms that should lead to narrow Feshbach resonances, suitable for magnetoassociation10,11,12. In this letter we describe the detection of Feshbach resonances in mixtures of 87Sr or 88Sr with 87Rb. The coupling between atomic and molecular states arises from two mechanisms previously predicted10,11,12 and an additional, weaker mechanism that we identify here.
The experimental signature of a Feshbach resonance is field-dependent loss of Rb atoms. This may arise from either three-body recombination or inelastic collisions, both of which are enhanced near a resonance. We perform loss spectroscopy using an ultracold Rb–Sr mixture, typically consisting of 5×104 Rb atoms mixed with 106 87Sr or 107 88Sr atoms at a temperature of 2 to 5 μK (see Methods). Figure 1 shows the observed loss features, eleven arising in the 87Rb–87Sr Bose–Fermi mixture and one in the 87Rb–88Sr Bose–Bose mixture. Ten loss features consist of a single, slightly asymmetrical dip with a full-width at half-maximum between 200 and 400 mG. The loss features labelled [1,0]a and [1,1]a each consist of several dips with a width of 20 to 60 mG at a spacing of 80 mG. We fit each dip with a Gaussian and give the resulting positions and widths in Table 1. None of these Rb loss features arises in the absence of Sr, proving that they depend on Rb–Sr interactions.
Both the atomic and molecular states are described by the total angular momentum of the Rb atom, f, and its projection m f onto the magnetic field. Where necessary, atomic and molecular quantum numbers are distinguished with the subscripts at and mol. In addition, the molecule has a vibrational quantum number n, counted down from n = −1 for the uppermost level, and a rotational quantum number L, with projection M L . 88Sr has nuclear spin iSr = 0, whereas 87Sr has iSr = 9/2 and a corresponding projection mi,Sr.
The Rb–Sr atom-pair states and the near-threshold molecular states lie almost parallel to the Rb atomic states as a function of magnetic field (see Supplementary Information). We can therefore use the Breit–Rabi formula of Rb for both the atom-pair states and the molecular states. This allows us to extract zero-field binding energies Eb of the molecular states responsible for the resonances, giving the values in Table 2. The crossing atomic and molecular levels are shown in Figs. 2 and 3, with filled symbols where we observe loss features.
To verify the bound-state energies and validate our model of Feshbach resonances, we use two-photon photoassociation spectroscopy. We detect the two n = −2 states (with L = 0 and 2) below the lower (f = 1) threshold of 87Rb–87Sr (states E and F in Table 2) at almost exactly the energies deduced from the resonance positions. All the states observed through Feshbach resonances (B to F) also arise to within 2 MHz in a more complete model of the Rb–Sr interaction potential, as described below.
Three different coupling mechanisms are responsible for the observed loss features. The first mechanism relies on the change of the Rb hyperfine splitting when the Rb electron distribution is perturbed by an approaching Sr atom and was proposed in ref. 10. Its coupling strength is proportional to the magnetic field in the field region explored here12. The coupling conserves m f and L and there are no crossings between atomic and molecular states with the same f and m f . This mechanism therefore produces Feshbach resonances only at crossings between atomic states with Rb with f = 1 and molecular states with f = 2. We observe one such resonance for each of 87Sr and 88Sr.
The second mechanism involves hyperfine coupling of the Sr nucleus to the valence electron of Rb and was first proposed in ref. 11. Since only fermionic 87Sr has a nuclear magnetic moment, this can occur only in Rb–87Sr collisions. This coupling conserves L and , with the selection rule . Crossings that fulfil these conditions occur also for molecular states with the same f value as the atomic state, which makes them much more abundant than crossings obeying the selection rules of the first mechanism. Feshbach resonances belonging to different mi,Sr are slightly shifted with respect to one another because of the weak Zeeman effect on the Sr nucleus and the weak Sr hyperfine splitting. However, since the shift is only ∼10 mG for neighbouring mi,Sr, much smaller than the width of the loss features of typically 300 mG, we do not resolve this splitting.
The third mechanism is the anisotropic interaction of the electron spin with the nucleus of either Rb or fermionic Sr. This mechanism can couple the s-wave atomic state to molecules with rotational quantum number L=2. As usual, the total angular momentum projection (now ) is conserved. If the Sr nucleus is involved, an additional selection rule is . By contrast, if the Rb nucleus is involved, the selection rule is . These loss features are made up of many (m f ,M L ) components, split by several hyperfine terms23; in some cases the components separate into groups for different values of M L . Three loss features are attributed to this mechanism and two of them ([1,1]a and [1,0]a) indeed show a structure of two or three dips.
To create a model of the Rb–Sr Feshbach resonance locations we start with a RbSr ground-state potential that we have previously determined by electronic structure calculations24. This will be described in detail in a future publication (A.C., manuscript in preparation). We carry out a three-parameter fit to adapt the model potential to match the molecular binding energies determined by two-photon photoassociation in three Rb–Sr mixtures (87Rb–84,87,88Sr), supplemented by binding energies determined from the Feshbach resonance positions. The parameters adjusted are two long-range coefficients, C6 and C8, and a short-range well depth. The molecular bound states obtained from the model are within 2 MHz of the measured values, and the resonance positions are within 2 G. Scattering calculations on this potential give interspecies scattering lengths and for 87Rb–87Sr and 87Rb–88Sr, where a0 is the Bohr radius. Our model also predicts the background scattering lengths and Feshbach resonance positions for all other isotopic Rb–Sr mixtures. For example, we predicted the position of the 87Rb–88Sr resonance after initially fitting the model only on photoassociation results for three isotopic mixtures and 87Rb–87Sr Feshbach resonances. This resonance was subsequently observed within 10 G of the prediction.
An understanding of resonance widths is crucial to molecule formation. However, the widths δ of the experimental loss features are dominated by thermal broadening, with relatively little contribution from the true resonance widths Δ. We obtain theoretical widths Δ from the Golden Rule approximation12 and include them in Table 1. Δ depends on the amplitude of the atomic scattering function at short range; it is largest when the background scattering length a is large, and is proportional to a in this regime12. This effect enhances all the resonance widths for 87Rb–87Sr.
In summary, we have observed Feshbach resonances in mixtures of Rb alkali and Sr alkaline-earth atoms. Similar resonances will be ubiquitous in mixtures of alkali atoms with closed-shell atoms, particularly when the closed-shell atom has a nuclear spin. Magnetoassociation using resonances of this type should be feasible (see Supplementary Information) and offers a path towards a new class of ultracold molecules, with electron spin and strong electric dipole moment. These molecules are expected to have important applications in quantum computation, many-body physics and tests of fundamental symmetries.
We prepare ultracold 87Rb–87Sr Bose–Fermi mixtures by methods similar to those in our previous work25. We transfer Rb and 88Sr from magneto-optical traps into a horizontal ‘reservoir’ dipole trap with a waist of 63(2) μm and a wavelength of 1070 nm. After Rb laser cooling we optically pump Rb into the f = 1 hyperfine state. By laser cooling Sr in the dipole trap on the narrow 1S0–3P1 line we sympathetically cool Rb. We then transfer between 5×104 and 1×105 Rb atoms into the crossed-beam ‘science’ dipole trap described below. We then ramp off the reservoir trap, discard 88Sr atoms and transfer between 1×106 and 2×106 87Sr atoms in a mixture of all ten nuclear spin states into the science trap. The final temperature is typically 2 to 5 μK. In order to prepare Rb in an equal mixture of all three f = 1 m f states we then randomize the distribution by non-adiabatic radiofrequency sweeps at a magnetic field of 130 G. To prepare Rb in the f = 2 hyperfine states we instead use optical pumping, which directly produces a nearly homogeneous distribution of Rb over the f = 2 m f states. To prepare 87Rb–88Sr Bose–Bose mixtures we do not discard 88Sr after transferring the gas into the science trap, and we skip the loading of 87Sr.
Science dipole trap
The science trap consists of two co-propagating horizontal beams and one vertical beam, all with coinciding foci. The first horizontal beam has a wavelength of 1064 nm and a waist of 313(16) μm (19(1) μm) in the horizontal (vertical) direction. The second horizontal beam has a wavelength of 532 nm and a waist of 219(4) μm (19(1) μm). The vertical beam has a wavelength of 1070 nm and a waist of 78(2) μm. The horizontal 1064-nm beam is typically used at a power of 5.7 W to 6.2 W and dominates the trap potential. The 532-nm beam is operated in the range 0.2 W to 0.4 W. The vertical beam is operated at 0.7(1) W to measure loss feature [1,−1]b and is off otherwise. These operating conditions result in typical trap depths of 40 μK × kB for Sr and 95 μK × kB for Rb, taking account of gravitational sag.
We observe Feshbach resonances through the field-dependent loss of Rb atoms. We submit the Rb–Sr mixture to a magnetic field of up to 55 G for a hold time of 1 to 10 s. Close to a Feshbach resonance, the rate of two-body inelastic collisions or three-body recombination is increased and atoms are lost. After the hold time we lower the magnetic field to near zero in 200 ms. During the next 10 ms, we ramp off the horizontal 532-nm beam and the vertical beam and decrease the power of the horizontal 1064-nm beam. This decrease lowers the evaporation threshold for Sr significantly, while Rb stays well trapped because its polarizability at 1064 nm is a factor of three higher. During the next 100 ms, Sr evaporates and cools Rb, which is advantageous for the subsequent imaging process.
At the end of the cooling stage, the science trap is switched off and a magnetic field gradient is applied to perform Stern–Gerlach separation of the Rb m f states. After 14 ms of expansion, absorption images of Sr and Rb are taken. To reduce sensitivity to Rb atom number fluctuations, we normalize the atom number of the Rb m f state of interest by the total atom number in m f states that are not lost. We verify that none of the loss features occurs in the absence of the Sr isotope concerned. For these verifications we need to retain a small amount of the other Sr isotope to allow sympathetic cooling.
Adaptation of experimental conditions
The width and depth of a loss feature depend on the measurement conditions. Thermal broadening sets a lower bound on the observable width, whereas hold time and peak densities affect the depth. For each resonance, we optimize the Sr density and hold time to maximize Rb loss without saturating the feature. We choose to use mixtures that contain much less Rb than Sr in order to obtain pronounced Rb loss features. Because of this atom number imbalance, and since most states can contribute to a given loss feature, the fractional Sr loss during the hold time is small, which keeps the Rb loss rate high.
The resonances that we attribute to coupling mechanism II, with the exception of [1,−1]a, are recorded under identical conditions. The hold time is 5 s and the peak densities for Rb and 87Sr are 2(1)×1011 cm−3 and 9(5)×1011 cm−3, respectively. The temperature of the Rb–Sr mixture before the hold time is 4.5(5) μK.
The [1,−1]a and [1,−1]c resonances exhibit higher loss rates. For these we use hold times of 1.5 and 1 s, respectively. The peak densities for Rb and 87Sr are 5(3)×1011 cm−3 and 2(1)×1012 cm−3. The temperature of the Rb–Sr mixture before the hold time is 3.0(1) μK.
The resonances that we attribute to mechanism III exhibit much lower loss rates. Therefore we use a hold time of 10 s. To measure feature [1,−1]b we also add the vertical trapping beam to increase the gas densities. The peak densities for Rb are 2(1)×1012 cm−3, 4(2)×1011 cm−3 and 3(2)×1011 cm−3 for features [1,−1]b, [1,0]a and [1, + 1]a, respectively. The peak densities for Sr are 5(2)×1012 cm−3, 2(1)×1012 cm−3 and 4(2)×1012 cm−3, respectively. The temperatures of the Rb–Sr mixture before the hold time are respectively 5(1) μK, 3.2(2) μK and 3.0(2) μK.
The [1, + 1] resonance observed in the Rb–88Sr mixture exhibits a high loss rate, because we typically load one order of magnitude more 88Sr atoms than 87Sr atoms into the science trap due to the naturally higher abundance of 88Sr. We use a hold time of 1 s and we do not use the 532-nm trapping beam. The peak densities for Rb and 88Sr are 6(3)×1011 cm−3 and 1.1(6)×1013 cm−3. The temperature of the Rb–Sr mixture before the hold time is 2.2(1) μK.
We use three pairs of coils to produce a homogeneous magnetic field across the atomic sample. The primary coils create a field up to ∼500 G with a resolution of 40 mG. These coils are used alone to record most of the loss features shown in Fig. 1. A second pair of coils is employed to resolve the substructure of the [1,0]a and [1, + 1]a loss features, with the primary coils producing bias fields of 278 G and 294 G, respectively. The secondary coils create a low magnetic field with a resolution of 3 mG. A third pair of coils is used to supplement the primary coils to observe the [1,-1]c loss feature, creating a bias field of 57 G.
We calibrate the magnetic field produced by the primary coils up to 290 G by spectroscopy on the narrow 1S0–3P1 line of 88Sr. We use a current transducer (LEM IT 600-S) to interpolate between the calibration points and to extrapolate to higher fields. We calibrate the secondary coils by recording one of the three [1, + 1]a loss features for different values of the field from the primary coils.
The magnetic field precision is limited by the resolution and noise of the power supplies. The inductances of the coils reduce the power supply noise contribution to less than 40 mG from all three pairs of coils combined, which is confirmed by the observation of a loss feature with a width of only δ = 24 mG. The accuracy is limited mainly by drifts in the ambient magnetic field. Monitoring the position of the 88Sr magneto-optical trap and the position of the loss feature due to a known Rb Feshbach resonance gives an upper bound of 350 mG for these drifts over the course of the present work. The calibration and statistical errors are typically one order of magnitude lower than the drifts. In addition, the positions of the loss maxima at finite temperature may differ from the positions of zero-energy Feshbach resonance positions. We account for this systematic error in the binding energies of Table 2 by adding an uncertainty of four times the root-mean-square width of the fitted Gaussian function.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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This project has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013) (Grant agreement No. 615117 QuantStro). B.P. thanks the NWO for funding through Veni grant No. 680-47-438. P.S.. thanks the National Science Center for support from grant 2017/25/B/ST4/01486. J.M.H. thanks the UK Engineering and Physical Sciences Research Council for support under Grant No. EP/P01058X/1.