Energy scaling of cold atom-atom-ion three-body recombination
We study three-body recombination of Ba + Rb + Rb in the mK regime where a single Ba ion in a Paul trap is immersed into a cloud of ultracold Rb atoms. We measure the energy dependence of the three-body rate coefficient and compare the results to the theoretical prediction, where is the collision energy. We find agreement if we assume that the non-thermal ion energy distribution is determined by at least two different micro-motion induced energy scales. Furthermore, using classical trajectory calculations we predict how the median binding energy of the formed molecules scales with the collision energy. Our studies give new insights into the kinetics of an ion immersed into an ultracold atom cloud and yield important prospects for atom-ion experiments targeting the s-wave regime.
When three atoms collide, a diatomic molecule can form in a three-body recombination (TBR) process. In cold neutral atomic gases, TBR was investigated for spin-polarized hydrogen as well as alkalis (see e.g. Hess et al. (1983); Burt et al. (1997); Esry et al. (1999)). In the context of Bose-Einstein condensation, TBR plays a crucial role as a main loss mechanism. By now, the scaling of TBR as a function of collision energy and scattering lengths in neutral ultracold gases has been investigated in detail D’Incao and Esry (2005). When considering TBR in atom-ion systems, one can expect three-body interactions to be more pronounced due to the underlying longer-range polarization potential. Energy scaling of TBR in charged gases was studied at temperatures down to a few K, especially for hydrogen and helium due to their relevance in plasmas and astrophysics (e.g. Krstić et al. (2003); Plašil et al. (2012)). Depending on the studied temperature range a variety of power laws was found but not a common threshold law. The recent development of novel hybrid traps for both laser cooled atoms and ions has opened the possibility to investigate cold atom-ion interactions and chemical reactions in the mK-regime and below. First experiments in such setups studied elastic and reactive two-body collisions (e.g. Grier et al. (2009); Zipkes et al. (2010); Schmid et al. (2010); Hall et al. (2011); Sullivan et al. (2012); Ravi et al. (2012); Sivarajah et al. (2012); Haze et al. (2013)). In accordance with the well-known Langevin theory, the corresponding reactive rates were measured to be independent of the collision energy Zipkes et al. (2010); Hall et al. (2011). Very recently we predicted a theoretical threshold law on the scaling properties for cold atom-atom-ion three-body collisions Pérez-Ríos and Greene (2015). Understanding the scaling of reaction rates with quantities such as the collision energy is crucial for fundamentally understanding TBR and for the prospects of the experimental realization of ultracold s-wave atom-ion collisions. Furthermore, as we will show here, studying TBR allows for insights into the kinetics of an ion immersed in a cloud of atoms. Experimentally, TBR in the mK regime was recently observed for Rb + Rb + Rb Härter et al. (2012) and Ba + Rb + Rb Krükow et al. (2016). In the Ba experiments TBR was already dominating over two-body reactions even for moderate atomic densities of cm.
This letter reports on the combined theoretical and experimental investigation of the energy scaling of three-body atom-atom-ion collisions in the mK regime. We measure the TBR rate coefficient of Ba in an ultracold Rb cloud as a function of the mean collision energy of the ion, , which we control via the excess micromotion (eMM) of the Paul trap. is formally distinguished from which is the TBR rate coefficient for a precise collision energy in the center of mass frame. By averaging over the ion energy distribution is obtained. We calculate using classical trajectory calculations (CTC) Pérez-Ríos et al. (2014); Pérez-Ríos and Greene (2015) and derive its energy scaling, . Agreement is found between theory and experiment if we assume that the energy distribution of the ion depends on multiple energy scales due to various sources of excess micromotion. Besides the prediction of , the CTC calculations also provide the binding energy distribution of the formed molecules and the scaling properties of these distributions when the collision energy is varied.
The experiments are performed in a hybrid apparatus that has already been described in detail elsewhere Schmid et al. (2012). After loading a single Ba ion by isotope selective, resonant two-photon ionization it is stored in a linear Paul trap driven at a frequency of with radial and axial trapping frequencies of , respectively. There, it is laser cooled to Doppler temperatures of mK. In order to perform our experiments in the electronic ground state we switch off the cooling and repumper light, before immersing the ion into the ultracold atomic cloud.
Once in the cloud, there is a complicated interplay of elastic two-body atom-ion collisions and the driven micromotion of the Paul trap. This interplay leads to a non-Maxwell-Boltzmann distribution of the ion’s kinetic energy DeVoe (2009); Zipkes et al. (2010); Cetina et al. (2012); Krych and Idziaszek (2015) with an equilibration time on the ms timescale 111The equilibration time can be estimated from the Langevin collision rate which at our given density is about kHz.. The average kinetic energy of the ion in the atom cloud is then determined by the available energy sources for the ion, such as the eMM energy Zipkes et al. (2010). In our experiment we can adjust by controlling one part of the eMM energy, , which is set via static electric fields. Concretely, we can write , where the offset energy contains all other energy contributions, e.g. phase micromotion (MM) Berkeland et al. (1998) or residual collisional effects Cetina et al. (2012); Krych and Idziaszek (2015). The proportionality factor , which depends on the atom-ion mass ratio and the trap parameters, is extracted from a MC calculation similar to Zipkes et al. (2011). We can tune accurately between 5K and 100mK. , on the other hand, is not known precisely. From independent measurements and MC calculations based on the scaling of elastic atom-ion collisions, we estimate to be in the range between 200 and 800 K.
The cloud consists of Rb atoms at a temperature of nK with a peak density of . It is cigar shaped with a radial and axial size of roughly m and m respectively. The atoms are spin polarized () and confined in a far off-resonant crossed optical dipole trap at a wavelength of 1064nm with a trap depth of K. We shift the ion into the cloud over a distance of m within 2 ms by changing the endcap voltage of the linear ion trap. After an interaction time of ms, during which the Ba ion is typically lost with a probability of up to 65 %, we separate the two traps again and detect whether the Ba ion is still present. For this, we shine a laser cooling beam focused to a waist of m through the Paul trap center and collect the possible fluorescence on a EMCCD camera for 100ms. If no Ba is detected, we conclude that a reaction must have taken place during 222 We note that our detection scheme cannot detect a reaction if the final product (e.g. after a secondary process) is again a cold Ba ion. From parallel experiments, however, we have no evidence for such a reaction outcome.
Repeating the single ion experiment roughly 170 times we extract the probability that Ba is still present. For the given experimental settings the ion loss is well described by an exponential decay of the form . This can be seen in the inset of Fig. 1, where we plot as a function of interaction time measured at K. Fig. 1 plots the loss rate as a function of . A Ba ion in our experiment is lost either by a two-body charge transfer or by a three-body event Krükow et al. (2016). The corresponding loss rate of the ion is . The charge transfer rate coefficient has been previously measured for Ba + Rb, /s (statistical and systematic errors in parentheses) Krükow et al. (2016) (see also Schmid et al. (2010); Hall et al. (2013)), and contributes less than Hz to the loss rate for the given atomic density. Also, it has been verified that is energy independent Grier et al. (2009); Zipkes et al. (2010); Hall et al. (2011), in consistency with Langevin theory. By subtracting this constant -loss from and dividing by the (constant) density we obtain (see Figs. 1 and 3b). Clearly, is energy dependent. As we will discuss later, we expect a scaling of with a power law, . Neglecting the atom motion due to ultracold temperatures we can express in terms of the ion kinetic energy , . We attempt to describe the scaling of the measured with a power law by fitting the expression
to the data. Here, and are free parameters. The constant /s is at and was determined in a parallel study Krükow et al. (2016). For the fit we discard data points above , as for such energies, the ion is not localized well enough in the center of the cloud. It probes areas of the atomic cloud at lower densities, hence decreasing the observed loss rate (see sketch in Fig. 1). The fit yields and K (green dashed line in Fig. 1), with the errors denoting a 1 statistical uncertainty of the fitted values. Interestingly, in our previous study of TBR of Rb + Rb + Rb Härter et al. (2012) we observed a similar scaling exponent of .
We now turn to investigate the scaling of TBR theoretically with a classical trajectory calculation (CTC) formalism. A classical treatment of the collision dynamics is appropriate, since the experiments described here in general involve much higher energies than the threshold energy of for entering the s-wave regime of Ba-Rb. We have adapted a recently developed method for the calculation of three-body recombination cross sections based on classical trajectories Pérez-Ríos et al. (2014); Pérez-Ríos and Greene (2015) for the study of atom-atom-ion recombination. The method employed relies on mapping the three-body problem into a 6-dimensional configuration space, described in hyperspherical coordinates, after separating out the center of mass motion Pérez-Ríos et al. (2014). Since the kinetic energy of the ion is typically several orders of magnitude higher than the temperature of the ultracold neutral atoms we fix one of the hyperangles associated to the ratio of the atom-ion versus the atom-atom initial momentum, guaranteeing that in the center of mass coordinate system 95 % of the collision energy is along the direction of the ion. In the classical trajectory calculations only Rb-Rb collisions in triplet states are considered and spin flip transitions are neglected. For the Rb-Rb pair interaction we employ the potential of Strauss et al. Strauss et al. (2010). On the other hand, the Ba-Rb interaction potential is taken to be , where 140 a.u. denotes the experimental long-range value of the interaction and represents the position of the minimum of the potential, taken from Ref. Krych et al. (2011).
The TBR rate for Ba + Rb + Rb has been computed by running trajectories per collision energy. We checked that during the simulation the total energy and angular momentum is conserved up to the fifth decimal place. Details about the numerical method employed to solve Hamilton’s equations of motion, in conjunction with the sampling of the initial conditions, can be found in Pérez-Ríos et al. (2014). Fig. 2a shows a three-body trajectory that results in a recombination event with a collision energy of 100 K. This particular trajectory leads to large size ( 800 a), very weakly-bound molecular ion. Counting the fraction of trajectories that lead to molecule formation we can extract the TBR rate coefficient for Ba + Rb + Rb. Fig. 2b plots as a function of collision energy . We compare these CTC calculations (diamonds) with an analytically derived scaling law Pérez-Ríos and Greene (2015) where (dashed line in Fig. 2b and find very good agreement.
Strikingly, the theory prediction of does not seem to agree well with the experimentally observed value of from the fit of Eq. 1 to our data. We explain this discrepancy as follows. In contrast to the theoretical approach where is determined for a precisely defined collision energy , in the experiments we observe , an average over a distribution of collision energies, calculated as
Here, is a list of the relevant energy scales that determine the distribution, such as the experimentally tuned or . We extract these distributions with a MC calculation based on Zipkes et al. (2011). If only a single scale is present, the energy distributions can be expressed as functions of the ratio ,
Fig. 3a shows three calculated distributions each with its own scale . The distributions for = 1mK (green) and 20mK (red) have the exact same shape, a consequence of Eq. 3. The third distribution mK (blue), generated with a phase micromotion has a somewhat different shape. Using Eq. 2 one can show that distributions which satisfy Eq. 3 translate the power law into . In our experiment, however, where at least two energy scales, and occur, this translation of the scaling breaks down and Eq. 1 cannot be used in the data analysis anymore. Instead, we calculate with Eq. 2 to directly compare theory and experiment. The choice and magnitude of is the only free model parameter. Here, we assume that is entirely determined by phase micromotion, . The phase micromotion is chosen to be shared equally between both pairs of opposing radio frequency (RF) driven electrodes Berkeland et al. (1998). Fig. 3b shows the experimental (full circles), together with the calculation (blue solid line) with K, 333Such a phase micromotion can be caused by a length difference of mm between the cables supplying opposing RF electrodes, which is well within the tolerances of our setup.. The shape of the theory curve describes the experimental data quite well, apart from an overall factor of about 1.1 (see blue and red solid lines). In general, the overall magnitude and energy dependence of is reproduced by the presented ab initio CTC treatment down to the mK-Regime, which is remarkable as is the only free parameter.
We now turn to briefly discuss the molecular products after TBR. In a previous study of TBR for He, it was suggested that the binding energy of the products is correlated with the collision energy Pérez-Ríos et al. (2014). We find again the same behavior for TBR of an ion with two atoms. Fig. 4a shows two logarithmically binned histograms of molecular binding energies after TBR. The maximum of each histogram can be considered the typical binding energy and is shown in Fig. 4b as a function of the collision energy . A fit to a power-law dependence gives for the energy range investigated here. Thus our calculations suggest that the formation of deeply bound molecules after TBR should be highly improbable at low collision energies.
The present CTC results also suggest that BaRb should be the dominant product state of the three-body recombination in the collision energy range considered here. Indeed, we have observed the formation of BaRb ions in our experiment. However, collisional or light induced secondary processes lead to short lifetimes. A detailed study of the initial TBR products and involved secondary reactions is currently in progress and needs to be discussed elsewhere.
In conclusion, we have investigated the energy scaling of three-body recombination in an atom-ion system down to mK energies. Single Ba ions in contact with ultracold Rb atoms have been used to measure the TBR rate coefficient . Utilizing classical trajectory calculations, we numerically accessed the TBR rate coefficient for the Ba + Rb + Rb system for various collision energies. We find a power law scaling of the form with an exponent . Our experimental and theoretical studies indicate that the presence of several energy scales gives rise to energy distributions of the immersed ion which impede a direct application of scaling laws to the measured data. The obtained energy scaling provides an important insight for prospects of atom-ion experiments in the ultracold regime, as the already strong TBR rate observed here will increase by another three orders of magnitude once the s-wave regime at nK is reached.
This work was supported by the German Research Foundation DFG within the SFB/TRR21 and by the Department of Energy, Office of Science, under Award Number DE-SC0010545. A.K. acknowledges support from the Carl Zeiss Foundation. J.P.-R. and C.H.G. thank Francis Robicheaux for many fruitful discussions. J.H.D. and C.H.G acknowledge inspiring interactions within program INT-14-1.
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