Collisions in a dual-species magneto-optical trap of molecules and atoms

We study inelastic collisions between CaF molecules and $^{87}$Rb atoms in a dual-species magneto-optical trap. The presence of atoms increases the loss rate of molecules from the trap. By measuring the loss rates and density distributions, we determine a collisional loss rate coefficient $k_{2} = (1.43 \pm 0.29) \times 10^{-10}$ cm$^{3}$/s at a temperature of 2.4 mK. We show that this is not substantially changed by light-induced collisions or by varying the populations of excited-state atoms and molecules. The observed loss rate is close to the universal rate expected in the presence of fast loss at short range, and can be explained by rotation-changing collisions in the ground electronic state.


Introduction
The study of collisions is an important topic in the field of cold atoms and molecules. Collisions are essential for sympathetic cooling, evaporative cooling and the association of atoms to form molecules. Low-temperature collisions are sensitive to the long-range part of the interaction potential and can be used to determine the energies of the highest-lying bound states of the potential. Inelastic and reactive collisions often limit the densities and lifetimes that can be reached with trapped atoms and molecules. Understanding such collisions is important and may suggest methods of controlling the collisions and avoiding the limitations that they otherwise impose.
At temperatures in the range 10 mK to 1 K, collisions have been studied using crossed and merged beams [1,2,3,4,5], and using mixtures loaded into traps by buffergas cooling or deceleration [6,7,8,9,10]. At lower temperatures, collisions between laser-cooled atoms have been studied extensively, both by theory and experiment [11]. More recently, collisions between ultracold molecules [12] have received considerable attention. These collisions result in rapid loss from optical traps, due either to chemical reactions or the formation of long-lived complexes that are excited by the trap laser [13,14,15,16,17]. Recently, elastic collisions between ultracold atoms and molecules in optical traps have been observed and used to bring the species into thermal equilibrium [18,19], and inelastic collisions between laser-cooled atoms and molecules have been studied in a magnetic trap [20].
In this paper, we present the first study of collisions between atoms and molecules in a magneto-optical trap (MOT). We create a dual-species MOT of 87 Rb and CaF and study the loss of molecules from the trap due to collisions with the atoms. We measure the loss rate coefficient, investigate which of the many possible processes are dominant, and interpret our findings in the context of simple theoretical models. The investigation of atom-atom collisions in MOTs has a long history [21,22,23,24,25,26,27]. In these atomic MOTs, collisional loss rate coefficients are found to vary over several orders of magnitude and often show extremely strong dependence on the MOT intensity and detuning. In some cases, rate coefficients exceeding 10 −10 cm 3 s −1 have been measured. At low laser intensities, where the excited-state fraction is small and the trap depth is low, the dominant collisional loss mechanism tends to be hyperfine-changing collisions between ground-state atoms that release enough energy to eject the atoms from the MOT. At higher intensities, processes where a ground-state atom collides with an excited-state atom become important. Fine-structure-changing collisions can occur, releasing the energy of the excited-state fine-structure interval, which is typically much larger than the trap depth. In addition, a colliding pair may absorb a photon at intermediate range; the resulting molecule may spontaneously emit a photon of lower energy at short range, with the energy difference converted into kinetic energy greater than the trap depth. This is known as radiative escape.
All the processes discussed above can also happen for the atom-molecule mixtures we study, but there can also be other processes. In particular, there can be rotationchanging collisions since laser cooling proceeds from the first rotationally excited state, and vibration-changing collisions since the MOT contains a mixture of molecules in a few vibrational states. In addition, a collision with an excited-state molecule can transfer the molecule to the opposite-parity component of the Λ doublet, which will eject it from the MOT. Molecules might also be lost due to light-induced processes analogous to radiative escape. Table 1   may be compared to the trap depth of the CaF MOT, which in the current experiments is U trap /h ≈ 1.6 GHz. In making this comparison, one should note that in a CaF-Rb collision that releases an energy E int much larger than the initial kinetic energies, the CaF molecule gains a kinetic energy of 0.6E int .   Figure 2 illustrates the setup we use to create the dual-species MOT of Rb and CaF. Coils inside the vacuum chamber create a quadrupole magnetic field with an axial gradient of 30 G cm −1 . A 3D MOT of 87 Rb atoms is formed using six independent laser beams. Each beam has a 1/e 2 radius of 1 cm and contains around 20 mW of cooling light red-detuned from the F = 2 → F = 3 transition, and 3 mW of repump light resonant with the F = 1 → F = 2 transition. The 3D MOT is loaded from a slow, collimated atomic beam extracted from a 2D MOT, which is formed in a glass cell attached to the main chamber. The scheme used to make the CaF MOT is the same as described in previous work, so is summarized only briefly here. A pulsed beam of CaF is produced by a cryogenic buffer gas source [28], slowed to low velocity using a frequency-chirped counter-propagating laser beam [29], and then captured into the MOT [30,31]. The six CaF MOT beams are generated by passing the same beam six times through the chamber. Dichroic mirrors are used to combine and separate the Rb and CaF light, and dichroic waveplates are used to generate the required circular polarizations.

Method
Each experiment begins by loading the desired number of atoms into the 3D MOT at a typical load rate of 3 × 10 9 atoms/s. We use between 9 × 10 8 and 8 × 10 9 atoms. Their temperature is 0.65 mK at the lower end of this range, rising to about 1 mK at the upper end. Once the atoms are loaded we switch off the 2D MOT and initiate the sequence to load the molecules, which takes 40 ms in total. We typically load about 10 4 molecules. While the CaF MOT is being loaded, I 00 is set to its maximum value of I max 00 = 1000 mW cm −2 . Then, unless stated otherwise, I 00 is ramped down to 0.2I max 00 ; this lowers the photon scattering rate, reduces the temperature of the molecules from 10 mK to 3.5 mK, and increases the MOT lifetime. At this intensity, the trap oscillation frequency is about 90 Hz and the trap depth is approximately 75 mK [31].
With both species trapped we acquire a set of 32 fluorescence images of the molecule cloud using a CCD camera. The images are separated in time by 18 ms, each has an exposure time of 10 ms, and the start of the first image defines t = 0. We also record a set of background images where everything is identical except that no molecules are loaded. After subtracting these background images from the signal images, we determine the number of molecules remaining as a function of time, t, and thus the loss rate from the MOT in the presence of the atoms, Γ 2 . Separately, we measure the loss rate of molecules in the absence of atoms, Γ 1 . In this case, the experimental sequence is identical except that the Rb repump light is turned off for both the 2D and 3D MOTs, so that no atoms are loaded. Γ 1 is determined by the rate of decay to states that lie outside the laser cooling scheme, and the rate of evaporation from the high energy tail of the distribution [31]. The difference between Γ 1 and Γ 2 is the atom-induced loss rate Γ Rb−CaF . Figure 3 shows the number of molecules as a function of time, normalized to the number at t = 0, both with and without atoms. We fit these data to a single exponential decay model with the rate as the only free parameter. For the data shown, where the density of atoms is (5.2 ± 1.0) × 10 10 cm −3 , we find Γ 1 = 6.39 s −1 and Γ 2 = 8.6 s −1 . For each experiment, these loss rates are measured 15 times, and this set of values is used to determine the mean value of Γ Rb−CaF , together with its standard error. The lifetime of the Rb MOT is around 8 s, which is at least 50 times longer than that of the CaF MOT, so the decay in the number of atoms during the measurement has a negligible effect on the loss rate measurements. In addition, the number density of atoms exceeds that of the molecules by 6 orders of magnitude, so loss of atoms due to collisions with molecules may safely be ignored.

Results
The collision-induced loss rate is related to the inelastic rate coefficient, k 2 , by where N Rb is the number of atoms and is the overlap integral between the density distributions of the two species, f s ( r), which are normalized such that f s ( r)d 3 r = 1. We measure N Rb by absorption imaging, as discussed in Appendix A. We measure the density distributions along y and z by imaging the fluorescence of both clouds onto the same CCD camera. We select the fluorescence from CaF using a band-pass filter centred near 606 nm. The fluorescence from Rb is much brighter, so we attenuate it using a neutral density filter. We find that the centres of the two clouds are displaced by 1.5 mm along y, but coincide along z. We attribute this displacement to imperfect polarization of all the beams, and an intensity imbalance between the counter-propagating CaF MOT beams. These imperfections are consequences of the optical setup (see Fig. 2), and are not easily corrected.
The circular points in Fig. 4 show measurements of Γ Rb−CaF as a function of N Rb . The data do not lie on a straight line because the overlap integral changes with N Rb . This can be seen in the lower panels of the figure, which show the density distributions along y for three different values of N Rb . These are measured for every N Rb , so the overlap integrals along y and z are known. We cannot measure the distributions along x since there is no optical access in this direction. Instead, we make the reasonable assumption that the MOTs are radially symmetric. This leaves a single undetermined parameter, ∆x, the displacement between the two clouds along x. Fitting to the data in Fig. 4 determines both k 2 and ∆x, as we now explain.  We fit two models to the data in Fig. 4. They differ in the method used to determine F. In the first method, which we call the analytical model, the clouds are described by Gaussian distributions in all three directions. In this case, Here, σ Rb i and σ CaF i are the rms widths and ∆i are the displacements. Fitting Gaussians to the measured distributions yields all these parameters at each value of N Rb , except ∆x. The values of σ CaF y,z show no dependence on N Rb . Neither do ∆y and ∆z, and we assume that the same is true for ∆x. It is convenient (though not necessary) to express σ Rb y,z as continuous functions of N Rb , and we find that a simple power-law model, σ Rb Rb describes the data well. In this way, we obtain F(N Rb ) with ∆x as the only free parameter. Fitting this model to the data in Fig. 4 yields the line shown in the figure, with best-fit parameters k 2 = (1.29 ± 0.21) × 10 −10 cm 3 /s and ∆x = 0.13 ± 0.07 cm.
For large atom clouds, the distributions deviate from Gaussians. Our second model, which we call the numerical model, does a better job of accounting for this. We express the overlap integral as a product of two parts, F = Υ 2D G. Here, Υ 2D is the overlap integral in 2D evaluated by multiplying the two images pixel by pixel, The sum is over all pixels, ∆A is the area in the object plane represented by one pixel, which is determined experimentally, and where p s j is the signal of species s in pixel j. In the x direction, we assume Gaussian distributions as before; G is their overlap integral. This model is defined only at the values of N Rb used in the measurement. The diamonds in Fig. 4 are the results of fitting this model to the data. The best-fit parameters are k 2 = (1.43 ± 0.29) × 10 −10 cm 3 /s and ∆x = 0.13 ± 0.07 cm. We take these results to be the more accurate ones, but note that they are entirely consistent with the ones found from the analytical model, showing that it makes little difference whether we use the actual distributions or Gaussian fits to those distributions.
Next, we turn to the question of the process responsible for the observed collisional loss. The dual-species MOT is a mixture of atoms in the ground ( 2 S) and excited ( 2 P 3/2 ) states with molecules in the ground ( 2 Σ) and excited ( 2 Π 1/2 ) states. The dominant inelastic process could involve any combination of these. There can also be light-induced processes due either to the light near-resonant with the atoms, or the light near-resonant with the molecules. Several hyperfine states are involved too. By measuring k 2 under various conditions, we disentangle the possible mechanisms. We independently modulate the intensities of the Rb and CaF MOT light, using a fixed modulation amplitude of 100%, a fixed frequency of 10 kHz, and a variable duty cycle, η s on , where s denotes the species. The modulation frequency is much faster than the trap oscillation frequencies (0.1-1 kHz) but much slower than the rate at which the state populations adjust to a change of intensity (> 1 MHz). The effective restoring force in the trap, the trap depth, the time-averaged excited-state fraction, and any light-induced inelastic rates, must all be proportional to η s on . Figure 5(a) shows how k 2 varies with the duty cycle of the Rb trap light, η Rb on . We use the method described above to determine k 2 in each case. The size of the Rb cloud changes with η Rb on . For example, the radial and axial rms radii of the cloud change from {σ y , σ z } = {1.78, 1.66} mm when η Rb on = 1 to {2.38, 2.2} mm when η Rb on = 0.1, with the largest change for duty cycles between 0.3 and 0.1. This change reduces the overlap between the clouds at low duty cycles, which in turn decreases Γ 2 , as can be seen in the inset of Fig. 5(a). The position of the cloud varies by less than 150 µm, while the number of atoms remains constant. Using the measured density distributions, we determine the overlap integral, F, for each value of η Rb on , using the analytical model.  Fitting these data to a linear model gives a gradient of (2.4±3.0)×10 −11 cm 3 /s, which is consistent with zero. The hypothesis that the observed loss is due entirely to collisions with excited-state atoms can be rejected with very high confidence since the intercept of the linear fit is 9.5σ away from zero. The hypothesis that the loss is due entirely to processes induced by the Rb cooling light can be rejected at the same level. If both ground-state and excited-state atoms are involved, we can write where k g 2 and k e 2 are the loss rate coefficients for collisions with ground-state and excited-state Rb atoms, and f e is the excited-state fraction whose value at η Rb on = 1 is f max e . The excited-state fraction in a Rb MOT has been studied experimentally over a wide range of parameters in Ref. [32]. For our parameters (detuning of −3Γ, intensity of 72.5 mW cm −2 ), the result is f max e 0.15. From the linear fit, we obtain k g 2 = (1.33 ± 0.14) × 10 −10 cm 3 s −1 and k e 2 = (2.9 ± 1.9) × 10 −10 cm 3 s −1 . Figure 5(b) shows how Γ 1 , Γ 2 and k 2 vary as we change the duty cycle of the CaF MOT light, η CaF on . The Rb MOT light is on continuously. In these experiments, the rms radii of the CaF cloud change from {σ y , σ z } = {1.40, 1.05} mm at η CaF on = 1 to {1.96, 1.28} mm at η CaF on = 0.3. The change in the cloud position is less than 0.2 mm and has no significant effect. As the duty cycle is lowered towards 0.6, both Γ 1 and Γ 2 decrease. This is due to the reduction in the time-averaged scattering rate. For duty cycles below 0.6 the loss rate increases again because the trap depth becomes too small to confine the higher-energy molecules. The linear fit to the k 2 data has a gradient of (7 ± 5) × 10 −11 cm 3 /s, which is consistent with zero. The intercept is 2σ away from zero, so the hypothesis that the observed loss is due entirely to collisions with excited-state molecules seems unlikely.
In addition to changing the duty cycles, we measure the loss rate coefficient over a wide range of CaF MOT light intensities, I 00 . Figure 5(c) shows Γ 1 , Γ 2 and k 2 for values of I 00 between I max 00 = 1000 mW/cm 2 and 0.05I max 00 . From the MOT fluorescence, we find that the excited-state population changes by a factor of 5 over this range. Lowering the intensity over this range reduces the geometric mean size of the cloud by 14% due to a decrease in temperature, and reduces the displacement between the two clouds by 0.6 mm. Fitting the data to a linear model, we find a gradient of (0.8 ± 2.7) × 10 −14 cm 5 /(s mW), showing that the trap light has no effect on the loss at this level of uncertainty. The intercept is 6.7σ from zero, so the hypothesis that the loss is due entirely to the presence of near-resonant light can be rejected with high confidence.

Discussion
In earlier work using CaF and Rb mixtures in a magnetic trap, a loss rate coefficient of k 2 = (6.6 ± 1.5) × 10 −11 cm 3 /s was measured for molecules in N = 1 at a relative temperature of 133 µK [20]. There was no significant dependence on the hyperfine state. This result was compared to the predictions of a single-channel model based on quantum defect theory [33], which takes account of quantum reflection off both centrifugal barriers and the long-range attractive potential. The model is characterized by two parameters, a short-range phase shift δ s and a short-range loss parameter y, which it assumes are independent of energy and partial wave. The measured loss rate was close to the universal rate [34], where every collision that reaches short range results in loss (y = 1). However, it was shown that such a rate can arise for values of y as low as 0.03 for certain values of δ s . The loss was attributed to rotational relaxation, which is driven by the anisotropy of the interaction potential. Such loss is expected to be fast (but not necessarily universal) when the anisotropy is large compared to the spacing between the rotational states. The kinetic energy released in a collision that changes N from 1 to 0 is 20.5 GHz, which is large compared to the trap depth.
For the present measurements, the effective temperature of the relative motion is T = µ(T Rb /m Rb +T CaF /m CaF ) = 2.4 mK, where T a and m a are the temperature and mass of species a, and µ is their reduced mass. At this temperature, the results of the singlechannel model have almost no dependence on δ s because shape resonances are washed out by thermal and partial-wave averaging [35]. The loss rate can be approximated as where the probability P of loss for collisions that reach short range is P = 4y/(1 + y) 2 .
Here k max 2 (T ) is the thermally averaged universal loss rate, which depends only on the collisional reduced mass µ and the asymptotic interaction potential −C 6 R −6 ; using the value of C 6 estimated in Ref. [36], we calculate k max 2 = 1.32×10 −10 cm 3 /s at T = 2.4 mK. Our experimental result is thus consistent with universal loss, as observed previously for rotation-changing collisions at lower temperature. The present experiment limits P to values greater than 0.86, corresponding to y > 0.46, and rules out the lower range of y permitted by the experiments at 133 µK.
At the temperature of the present experiment, where 4 to 7 partial waves contribute, universal loss is well approximated by a Langevin capture model [33]. In this model, all classical trajectories with sufficient energy to cross the centrifugal barrier are assumed to result in loss. For an asymptotic potential −C 6 R −6 , the Langevin model gives [37,38] k max 2 (T ) = 2 11/6 Γ(2/3) π µ where Γ(z) is the gamma function. For Rb+CaF at T = 2.4 mK, this evaluates to k max 2 = 1.34 × 10 −10 cm 3 /s, in good agreement with both the experimental value and the quantum-mechanical universal loss rate. The Langevin rate scales as T 1/6 ; in the case of universal loss, this scaling remains valid down to temperatures around T ∼ E 6 /k B [33], ‡ where E 6 = 3 / 8µ 3 C 6 , which is 124 µK for Rb+CaF. The ratio between the values of k 2 measured at 2.4 mK and 133 µK is consistent with this scaling.
It is interesting to compare our results to those found for atomic MOTs [21,22,23,24,25,26,27]. Hyperfine-changing collisions between ground-state atoms are an important loss mechanism in atomic MOTs when the trap depth is smaller than the energy released in such collisions. Loss rate coefficients comparable to those measured here have been observed. As the trap depth increases, this mechanism is suppressed and the loss rate can be reduced by orders of magnitude. In our MOT, collisions that change only the hyperfine state of the molecule do not release enough energy for the molecule to be lost. However, in a low-energy collision that changes the hyperfine state of 87 Rb, the molecule emerges with a velocity of 7.4 m/s. This is close to the capture velocity of the MOT at high intensity, which is about 11 m/s [39]. The trap depth decreases as the duty cycle is lowered, so we might have expected a strong dependence on the duty cycle, as loss due to hyperfine-changing collisions turns on. However, as argued above, rotational relaxation causes loss that is already at the universal limit. In this limit, adding an additional collisional loss mechanism, no matter how strong, does not increase the loss rate. The same argument applies to radiative escape, where a colliding pair absorbs a photon at intermediate range and emits a lower-energy photon at short range. ‡ In the nonuniversal case, when y 1, T 1/6 scaling can break down even at temperatures well above E 6 /k B [33].
Excited-state atoms and molecules form distinct populations with their own loss mechanisms and their own distinct value of the universal loss rate. In particular, there can be loss due to fine-structure-changing collisions with excited-state atoms, which is known to be important in atomic MOTs, and due to Λ-doublet-changing collisions involving excited-state molecules. The molecules are excited to the positive-parity component of a Λ doublet, which lies 1.36 GHz higher in energy than the negativeparity component. A collision that changes this parity releases the energy and takes the molecule out of the laser-cooling cycle. Such transitions are driven directly by oddorder terms in the Legendre expansion of the anisotropy of the interaction potential [40]. These terms are large compared to the parity splitting, so are also expected to cause fast loss, which may be sufficient to reach the universal limit. The excited-state universal rate will differ from that for the ground state due to the different value of C 6 . However, the difference is likely to be small because the Langevin rate depends only weakly on C 6 , as C 1/3 6 . The observed insensitivity of the loss rate to the CaF excited-state fraction is consistent with parity-changing collisions that are sufficient to cause universal loss.

Conclusions
We have developed a dual-species magneto-optical trap of CaF molecules and Rb atoms and measured the loss rate of CaF from the trap due to inelastic collisions with the atoms. At an effective collision temperature of 2.4 mK, the measured loss rate coefficient is k 2 = (1.43 ± 0.29) × 10 −10 cm 3 /s. This result is consistent with the universal loss rate and with the loss rate calculated in the classical limit using a Langevin capture model. The result constrains the probability of loss for collisions that reach short range to P > 0.86. The corresponding constraint on the loss parameter is y > 0.46. We expect y to be independent of energy, so this constraint should apply also to collisions in the ultracold regime. Rotational relaxation of ground-state molecules colliding with ground-state atoms is sufficient to account for the loss rate observed. We cannot rule out contributions from excited-state atoms or molecules, or from light-induced processes, but these processes alone are insufficient to explain our observations.
Our dual-species MOT could be used as a starting point for forming the triatomic CaFRb molecule. The formation of diatomic molecules by photoassociation of atoms is often studied by adding an additional laser to a dual-species MOT and measuring resonances in the loss rate as the frequency of this laser is scanned. This is unlikely to work in our dual-species MOT since the loss rate is already close to the universal limit; adding a new loss mechanism cannot much increase the observed loss rate. However, the atoms and molecules could be loaded from the MOT into a conservative trap where molecule formation by photoassociation or magnetoassociation through a Feshbach resonance could be studied.
Our results are also important for designing an experimental protocol for sympathetic cooling of molecules using evaporatively cooled atoms [36]. Again, the atoms and molecules first need to be loaded from the MOT into a conservative trap.
The rapid collisional losses observed in the dual-species MOT will limit the density that can be achieved. This limitation could be avoided by loading molecules and atoms sequentially into the conservative trap. Collisional losses in a magnetic trap are much slower, provided that the molecules are in the rotational ground state [20]. Laser cooling is now being applied to many molecular species (see [41] for a recent review) and the methods developed here could be used to produce a wide range of atom-molecule mixtures and to study how they interact at low temperatures.  Figure A1. Absorption of the probe beam by the atom cloud released from the MOT. In (a) the probe beam is switched on 1 ms after turning off the MOT coils and kept on for a variable time t p before an absorption image is acquired. In (b) the absorption is measured for different delay times t d after turning off the coils. The solid lines in both figures are fits to exponential models.
turning off the MOT coils. Figure A1(b) shows how the total absorption of the probe beam varies with the delay time t d between switching off the magnetic field and acquiring the absorption image. The absorption is highest at t d = 0, presumably because the MOT magnetic field tends to shift the atoms closer to resonance with the light. Fitting an exponential model to these data gives a time constant of 0.73(9) ms. In the collision experiments, we acquire images at t d = 1 ms so as to avoid excessive expansion of the atom cloud. The absorption is about 19% higher at this time compared to long times when the magnetic field has decayed away and we account for this when we estimate the number of atoms.