Collisional and molecular spectroscopy in an ultracold Bose-Bose mixture

The route toward a Bose-Einstein condensate of dipolar molecules requires the ability to efficiently associate dimers of different chemical species and transfer them to the stable rovibrational ground state. Here, we report on recent spectroscopic measurements of two weakly bound molecular levels and newly observed narrow d-wave Feshbach resonances. The data are used to improve the collisional model for the Bose-Bose mixture 41K87Rb, among the most promising candidates to create a molecular dipolar BEC.


Introduction
A new tide in the domain of quantum degenerate gases is rising. Degenerate dipolar molecules, created from ultracold or degenerate atoms, are within reach. Such molecules will enable the study of zero-temperature systems with strong long-range interactions, whereas degenerate atoms interact substantially only through contact potentials. Degenerate molecules with dipole-dipole interactions will provide new quantum phases, will allow the simulation of magnetic spin systems and will provide candidate qubits for quantum computation.
Ultracold molecules have been produced by several groups, using either laser cooled atoms in a magneto-optical trap (MOT) [1] or degenerate atoms [2]. Due to the energymomentum conservation, molecular association cannot be a simple two-body process but requires a three-body collision, the exchange of photons (photoassociation) or adiabatic transitions (magnetoassociation).
Molecular Bose-Einstein condensates have already been created [3,4], but so far only with homonuclear dimers, whose electric dipole moment is necessarily zero. On the other hand, heteronuclear molecules have been created starting from ultracold but not degenerate samples [5]. A common challenge to all weakly bound dimer samples is relaxation decay. While in a MOT production of molecules directly in the rovibrational ground state has been demonstrated [6], the more efficient association from degenerate or quasi degenerate (T < 1µK) atomic samples yields molecules in weakly bound state that are inevitably unstable to relaxation toward lower lying rovibrational levels. Only recently have 40 K 87 Rb dimers, which were created by magnetoassociation, been transferred to the rovibrational ground state. Their electric dipole moment has been measured to be 0.566 Debye [7], in an experiment which represents a milestone in the route toward degenerate dipolar molecules.
The production of a Bose-Einstein condensate of dipolar molecules, however, requires association of either two bosons or two fermions. Therefore we chose to investigate the Bose-Bose mixture 41 K 87 Rb in which we associated the first doublespecies bosonic molecules [8]. Later, bosonic dimers were also created in a Fermi-Fermi mixture of 6 Li 40 K [9]. For both association and state-transfer, an accurate knowledge of the interatomic potential is essential. In this work we report spectroscopic measurements on the 41 K 87 Rb weakly bound levels, together with newly observed d-wave Feshbach resonances. We use this new set of data to improve the collisional model for KRb [10], earlier adjusted to fit the extensive observations of Feshbach resonances in the isotopic 40 K 87 Rb mixture [11,12]. An accurate knowledge of the interatomic potential between 41 K and 87 Rb is essential for devising the transfer of molecules towards the low rovibrational states. The present work is therefore instrumental to the production of bosonic KRb molecules with long-range dipolar interactions.
The paper is organized as follows: in section 2 we describe our experimental procedure, in section 3 we present the data. The theoretical results obtained by the adjustment of the collisional model are reported in section 4. Finally we discuss the prospects of a KRb dipolar condensate.

Experimental setup
Our experimental setup consists of two separate 2-dimensional magnetooptical traps (MOT), that deliver cold atomic beams of 41 K and 87 Rb [13] into a double-species 3-dimensional MOT. The laser cooled mixture is loaded in a quadrupole magnetic trap with a gradient of 260 G/cm that is subsequently translated by 31 mm with a motorized stage. The quadrupole magnetic field is converted into a harmonic magnetic trap generated by means of the millimetric trap described in [14]. Then we start the microwave evaporation that expels only 87 Rb atoms and cools the mixture (sympathetic cooling) down to approximately 2 µK. At this stage, the magnetic trap is replaced by a crossed dipole trap generated by two orthogonal laser beams at 1064 nm, with waists of ∼ 90 µm. Once the magnetic potential is completely extinguished, both species are transferred from the |F = 2, m F = 2 to the |1, 1 hyperfine state, by means of consecutive radio-frequency (rf) ramps in presence of an horizontal bias magnetic field of 7 G (adiabatic rapid passage). To guarantee the stability of the mixture against spinchanging collisions, it is important to transfer the species with the largest hyperfine splitting first and only later the second species: in our case we transfer 87 Rb first and then 41 K. The adiabatic rapid passages last approximately 30 ms each and feature efficiencies comprised from 80 to 95%.
We raise a homogeneous magnetic field (Feshbach field) along the vertical direction to access the interspecies Feshbach resonances. We calibrate the Feshbach field by measuring the frequency of 87 Rb hyperfine transitions. While the field resolution is 20 mG, reproducibility is limited to 50 mG.
The mixture is further cooled by lowering the power of the dipole trap beams to reach temperatures ranging from 300 to 600 nK with typically a few 10 4 atoms of each species. The mixture is now prepared for the subsequent experiments.
To associate the molecules and to measure the dimer binding energy, we modulate the Feshbach field with an additional excitation coil, driving transitions from unbound to bound pairs. The excitation frequencies range from 50 to 200 kHz, but we can drive the transitions also with half the resonant frequency, effectively doubling the range of the measurable binding energies. The excitation is 15 to 30 ms long, with a square amplitude envelope. As explained in Ref. [8], we measure our typical excitation amplitudes to be 130 mG.
The association of molecules is revealed by the loss of trapped atoms after the excitation pulse as we scan the excitation frequency. Such losses, resonant with the excitation frequency, occur when unbound pairs are converted into weakly bound dimers. Indeed these dimers are lost from the trap as soon as the subsequent inelastic collisions rapidly drive them into more deeply bound levels with a kinetic energy larger than the trap depth.
Likewise, the detection of Feshbach resonances is obtained by the atomic losses caused by three-body recombination collisions. We therefore measure the number of atoms remaining after a fixed time in the dipole trap (hold time), as we scan the Feshbach field [15]. This method is especially useful for narrow resonances (such as those of higher partial waves), while it is poorly accurate in case of broad Feshbach resonances, since the complex dynamics of atoms and weakly bound dimers must be taken into account to determine the exact position of Feshbach resonance from the inelastic losses. When possible, the positions of Feshbach resonances are better determined from extrapolation to zero of the measured binding energies.

Experimental results
We report here two different results which are both important for improving the accuracy of the collisional model for 41 K 87 Rb, namely the measurement of the binding energies of the s-wave Feshbach dimers and the detection of two narrow d-wave Feshbach resonances.

s-wave molecular levels
In the vicinity of Feshbach resonances, the binding energy of the shallowest molecular levels can be accurately measured using rf spectroscopy. Unbound atom pairs are associated into weakly bound dimers by means of an oscillating magnetic field b cos(ωt). In our experiment, the b field is parallel to the Feshbach field, therefore it represents a modulation of the latter [16]. The association occurs resonantly when the oscillation frequency matches the energy difference between the unbound pair, E = E kin , and the molecular level, The associated dimers undergo inelastic collisions with the unbound atoms and decay to more deeply bound levels ("relaxation collisions"). In this process the binding energy is converted into kinetic energy that is shared by the collisional partners, which are both expelled from the trap. Thus, association of molecules can be detected as a resonant loss of the atomic sample, even if the molecules themselves are too short lived to be directly detected.
In [8] we reported the first creation of heteronuclear bosonic molecules by means of rf association. We measured the binding energies of the Feshbach molecular levels next to the Feshbach resonances at 38 and 79 G. The atomic losses display a line shape, shown in figure 3.1, that must be modelled to accurately extract the binding energy. For this purpose, we employ a simple model, which is described in the Appendix, that captures the salient features of the line shape. Our analysis reveals that the line shape is inhomogeneously broadened due to the thermal distribution of kinetic energies of the unbound atoms. The uncertainty of the measured resonant frequencies is 1%, which is mainly due to the fit precision.
In the following, we use these data to improve on the collisional model which predicts the weakly bound molecular levels near the scattering threshold and the scattering length as a function of the magnetic field.

Higher order Feshbach resonances
In addition to the strong losses caused by two s-wave Feshbach resonances, at 38 and 79 G respectively, we observe two narrow features, that we attribute to higher partial waves Feshbach resonances based on the collisional model developed in [10]. These peaks, shown in figure 2, were observed by simply holding the atomic mixture in the optical trap for 100 ms at different magnetic fields. A first loss feature, detected at a temperature of 400 nK, is centred at a magnetic field of 44.58(0.05) G, with a very narrow half-width at half-maximum (HWHM) of 0.1 G.
The second loss feature is detected at a temperature of 450 nK, centred at a magnetic field of 47.96(0.02) G. The width of this peak, HWHM=0.08 G, is at the limit of our experimental resolution. For both loss peaks the uncertainty on the position is mainly systematic and equals 0.05 G. The assignment of these narrow loss features to d-wave Feshbach resonances follows from the collisional model, as described in the next section.

Theoretical results
Collisional properties of KRb isotopes are well understood. In Ref. [10], data on 39 K 87 Rb and 40 K 87 Rb Feshbach resonances have been used to construct an accurate quantum scattering model. Here we add to the data set binding energies as a function of magnetic field of two near dissociation levels and magnetic resonance positions in the 41 K 87 Rb isotopic pair (see table 1 and table 2). Molecular state energies are particularly valuable as they are immune to possible systematic shifts affecting the determination of the two-body resonance locations obtained from the observed maxima of the three-body recombination rate.
Our numerical calculations use the collision model built in Ref. [10]. The shortrange molecular potentials are parameterized in terms of singlet and triplet s-wave scattering lengths, a s and a t . The long range interatomic interaction is expanded in a multipole series in terms of C 6 , C 8 and C 10 dispersion coefficients. Relativistic Table 2. Comparison of observed E m,exp and numerically calculated E m,th s-wave molecular levels as a function of the magnetic field B. The rightmost column shows the molecular quantum numbers (f m f ℓ ′ ). Errors in the E m,exp column have been used for the weighted χ 2 and account for direct energy error plus the uncertainty 0.05 G in the magnetic field (see text). Errors on theoretical data have been recalculated from the model.  (9) interactions are relatively weak for alkali species. Inclusion of the the dipolar interaction between the atomic electron spins and of the second-order spin-orbit interaction [17] is indeed sufficient to explain the available data on the isotopic pairs. Here, we vary the singlet and triplet s-wave scattering lengths and the van der Waals coefficient C 6 until a good agreement with the data is found. The additional potential parameters are kept fixed to the values of Ref. [10]. The energy levels included in the data are known to be molecular states with null orbital angular momentum ℓ ′ . The magnetic field location at which these levels become degenerate with the energy of the separated atoms corresponds to the two s-wave resonances observed in [15].
Note that in our fitting minimization procedure the magnetic field is the independent variable and it assumed without error. The actual experimental uncertainty on B is accounted for by projecting it on the energy axis and adding it in quadrature to the direct measurement. As it can be remarked from the data, the level corresponding to the 79 G resonance varies rapidly with magnetic field, such that a small B uncertainty of 0.05 G amounts to a ∼ 15 kHz error on the molecular state energy. Conversely, the level associated with the 38 G resonance varies more slowly with B and the error is determined by the direct energy measurement only.
We include in the data set two newly observed narrow features and the narrow s-wave resonance of [8], whose position can be very precisely determined. Magnetic resonances of different nature have been observed in cold atom collisions in a number of homonuclear [18] and heteronuclear systems [11,19]. In general, broad resonances result from spin-exchange coupling of incoming s-wave atomic pair with ℓ ′ = 0 molecules. Such processes are induced by the spherically symmetric exchange interaction and conserve the orbital angular momentum ℓ, its projection m and the projection m f of the total hyperfine spin of the system in the direction of the applied magnetic field.
Relativistic spin-spin and second order spin-orbit interactions are anisotropic and enable first order coupling of ℓ-wave atomic pairs with ℓ ′ symmetry molecules, provided that |ℓ ′ − ℓ| ≤ 2, the ℓ = 0 → ℓ ′ = 0 transitions being forbidden. Rotational symmetry about the external magnetic field implies exact conservation of the total angular momentum projection m + m f . The total interaction couples incoming s-wave atomic pairs with m f = 2 to states characterized by ℓ ′ = 0, 2 and 0 ≤ m f ≤ 4. Higher order couplings to different ℓ ′ and m f are possible but only give rise to much narrower features.
Collisions for atoms incoming in ℓ > 0 partial waves tend to be suppressed at temperatures below the ℓ-wave centrifugal barrier. We take into account spin-exchange p-wave resonances, characterized by ℓ ′ = 1 and m f = 2. Such resonances have been observed even at very low temperature [20]. Finally, p-wave resonances induced by spin-spin coupling are in principle possible but they can be expected to be even weaker because of the weak coupling strength and the energy suppression. The corresponding m f for direct coupling with incoming p-waves is bounded to 0 ≤ m f ≤ 4.
The model of Ref. [10] allows us to give a tentative assignment of the data. Towards this aim, it is instructive to calculate the near-threshold molecular levels for the lowest values of orbital angular momentum ℓ ′ = 0, 1, 2 (see figure 3). A simplified approach that neglects the spin interactions is fully adequate for this purpose, since the resulting level shifts are small. In this approximation m f , ℓ and m are good quantum numbers. The f quantum number is precisely defined only when B = 0. Thus, we use it as an approximate quantum label for the levels in the low-field domain B < 80 G of figure 3. We only show levels coupled in first order to incoming sor p-wave atoms. Figure 3 allows us to identify the molecular levels associated with the two observed loss features of figure 2. The one at lower field is univocally associated with a ℓ ′ = 2 molecular level. The only possible ambiguity for the higher field feature arises from the crossing of a ℓ ′ = 1 level with m f = 3 very close to the observed value. This possibility is excluded since this feature would be a spin-spin induced p-wave resonance. If such a weak feature were observable, one should then also observe a stronger spin-exchange p-wave loss feature at approximately 50 G, which is not detected. Moreover, with the present resolution one should observe the typical doublet structure of p-wave peaks [21]. We conclude that the molecular level associated with the 47 G resonance has also ℓ ′ = 2 angular momentum.
With this assignment, we optimize our collisional model to obtain the best-fit parameters: where a 0 denotes the Bohr radius and E h the Hartree energy. The reduced weighted χ 2 is 0.92 and the maximum deviation between theoretical and empirical values is about two standard deviations. The present values are in agreement with Ref [10]. In fact, if the bound states and the resonances in the three isotopic pairs are simultaneously fit, the following model potential parameters are determined: a s = − 109.6(2)a 0 a t = − 213.6(4)a 0 At variance with [10] the strength A ex of the exchange interaction [22] is also included in the fit parameters. The quality of the fit is similar to the one of [10] with a reduced χ 2 of 1.1 and a maximum discrepancy with the experimental data of at most two standard deviations. The theoretical data do not show any systematic positive or negative shift with respect to the observed features. This strengthens the conclusion obtained in Ref. [10] that breakdown of the Born-Oppenheimer approximation [23] does not produce measurable effects at the current level of precision.

Conclusions
We have shown that recently published data on rf association of Feshbach dimers and newly determined Feshbach resonances in higher order partial waves lead to an improved collisional model for 41 K 87 Rb. Such a collisional model is instrumental for future experiments on this mixture. In particular, KRb dimers appear among the best candidates to produce a molecular BEC with dipolar interactions. Indeed, bosonic KRb should be amenable to efficient transfer toward the rovibrational ground state with a single Raman pulse, following the pioneering demonstration on their fermionic 40 K 87 Rb counterparts [7]. The route to stable bosonic molecules presumably will take advantage of dimer association starting from atoms trapped in optical lattices. As already recognized in [24], a double Mott insulator with unit filling (per species) will enable the creation of molecules secluded in individual lattice sites, thereby shielded from collisions with both unbound atoms and other molecules [25]. In addition, the lattice is an essential tool for implementing several models of bosons with long range interactions [26]. time-dependent free hamiltonian isĤ 0 = (−E bind (B(t)) − iγ/2)m †m + p 2 /(2m A )â †â + p 2 /(2m B )b †b , where m A and m B denote the atomic masses and the time-dependence of the binding energy is brought by the dependence of the (positive-defined) binding energy E bind on the oscillating Feshbach field B(t) = B dc + b sin(ωt). The molecules have a finite lifetime γ −1 , due to relaxation processes. The coupling is due to the oscillating part of the Feshbach field. Following Ref. [27], we consider a single-particle coupling strength proportional to the derivative of the oscillating field Ω(B(t)) = Ω 0 (B dc ) cos(ωt), with Ω 0 (B) : Thus the rf coupling hamiltonian is given byĤ rf = Ω(t)(m †âb +â †b †m ). With this, we write the Heisenberg equation for the operatorsâ,b,m and take the expectation values while neglecting all correlations. In practice, we replace the above operators with c-numbers α, β, µ: As expected from the conservation of total number of particles, the quantity 2|µ| 2 + |α| 2 + |β| 2 is constant for γ = 0.
We can solve analytically this set of non-linear Bloch equations, provided we introduce (i) the rotating-wave approximation, i.e., we write the equations in terms of the slowly varying amplitudesα := α exp(ip 2 /(2hm A )t),β := β exp(ip 2 /(2hm B )t),μ := µ exp(−i(ω − E kin /h)t), with E kin = p 2 /(2m A ) + p 2 /(2m B ), and neglect all rapidly oscillating terms; (ii) the adiabatic approximation, whereby we consider that, since the molecular decay rate is fast, we can take the molecular amplitudeμ to adiabatically follow the evolution dictated by the slower rf coupling, in practice dμ/dt = 0. As a consequence, we have: Obviously the difference n := N A − N B stays constant, while the sum N tot := N A + N B obeys the equatioṅ N tot = −γ eff (N 2 tot − n 2 ), γ eff := γ that has the following solution For n = 0, the above reduces to N tot (t) = N tot (0) 1 + N tot (0)γ eff t (7) which is the well-known solution of the rate equationṄ = −γ eff N 2 , that is used to describe losses due to two-body collisions. The solution (6) depends on the atomic kinetic energy E kin , whose values are spread over a range of the order of the temperature T . Under our experimental conditions, the thermal distribution of kinetic energies dominates the broadening of the rf spectra over the linewidth dictated by the decay rate γ. In order to compare with the measured spectra, we need to take a thermal average of Equation (6) by convolving with the Boltzmann distribution of kinetic energies proportional to √ E kin exp(−E kin /k B T ). The resulting line shape describes well the strong asymmetry of the observed spectra.
A more detailed inspection of Equations (3) reveals another interesting feature: Molecular association also occurs for fractional frequencies of the resonant frequency (see figure 3.1). This is experimentally observed and confirmed by the results of numerical integration of Equations (3).
Indeed the fact that the binding energy is modulated at the rf frequency ω makes it possible that the molecular amplitude contains Fourier components oscillating at integer multiples of ω. When one of these harmonics is close to the time-averaged value of the binding energy, the transfer of population to the molecular level occurs.
Another important experimental finding confirmed by the numerical simulations is the shift of the resonant peak frequency occurring when the binding energy depends nonlinearly on the magnetic field detuning from the Feshbach resonance. This is also easily understood when we consider that the time-averaged binding energy is shifted from the value in the absence of modulation. If the binding energy is quadratic with the magnetic field detuning E bind = η∆B 2 , which is the case next to a Feshbach resonance, we have that E bind t = E bind (∆B dc ) + ηb 2 /2, since ∆B(t) = ∆B dc + b sin(ωt). This relationship has been experimentally verified [8].