Transport dynamics in a high-brightness magneto-optical-trap Li ion source

Laser-cooled gases offer an alternative to tip-based methods for generating high brightness ion beams for focused ion beam applications. These sources produce ions by photoionization of ultracold neutral atoms, where the narrow velocity distribution associated with microkelvin-level temperatures results in a very low emittance, highbrightness ion beam. In a magneto-optical-trap-based ion source, the brightness is ultimately limited by the transport of cold neutral atoms, which restricts the current that can be extracted from the ion-generating volume. We explore the dynamics of this transport in a Li magneto-optical trap ion source (MOTIS) by performing timedependent measurements of the depletion and refilling of the ionization volume in a pulsed source. An analytic microscopic model for the transport is developed, and this model shows excellent agreement with the measured results.


I. INTRODUCTION
Photoionization of laser-cooled atoms offers a novel pathway for constructing high brightness ion sources. [1][2][3][4][5][6][7] These sources are fundamentally enabled by the microkelvin temperatures achievable through laser cooling, where atomic ensembles are produced with a very small momentum spread. Upon photoionization and extraction, the low momentum spread in the transverse direction results in a very low emittance ion beam. This emittance, when combined with a reasonable amount of current, results in a brightness that rivals or surpasses conventional ion sources.
Cold atom ion sources benefit from several advantages over conventional tipbased ion sources, such as access to new ionic species, inherent isotopic purity, insensitivity to contamination, and a low energy spread, which reduces chromatic aberration in ion optical systems and enables source operation at low accelerating voltages. Applications for these new sources are broad, encompassing milling, imaging, spectroscopy, and implantation. For example, a Li magneto-optical trap ion source (MOTIS) has demonstrated high surface sensitivity in imaging, 8 imaging of optical modes in nanophotonic resonators, 9,10 and utility in studying Li-ion-battery-relevant materials at the nanoscale. 11,12 Cold atom ion sources of heavier species such as Cs enable nanometer scale milling resolution and improved secondary ion mass spectroscopy. 13 Recent work has demonstrated active feedback control of an ion beam using a measurement of the momentum of the photoemitted electron to correct the trajectory of each generated ion. 14,15 Such control raises the possibility of deterministic implantation of single ions, as well as active compensation of the random transverse energy of the beam to further increase the resolution of these instruments.
Several methods exist for laser cooling atoms into cold atomic beams or trapped gases to realize an ion source. The technical complexity and utility of each approach varies with atomic species, as does the path toward optimizing ion source brightness. In the case of Cs, an atomic beam source with multistage laser cooling has been demonstrated with peak brightness as high as 2.4×10 7 A m -2 sr -1 eV -1 , a value 24 times greater than the typical Ga + liquid metal ion source (LMIS) brightness. 13 Li, on the other hand, presents significant complexities for multistage laser cooling due to its high Doppler temperature and repumping requirements, and it is more practical to create and optimize a source on a simpler platform using photoionization in a magneto-optical trap (MOT). Although magneto-optical trap ion source (MOTIS) brightness is fundamentally limited by transport of cold atoms into the ionization volume, reasonably high brightness can nevertheless be achieved with Li. For example, a Li MOTIS has been optimized to have a peak brightness of 1.2×10 5 A m -2 sr -1 eV -1 and current as high as 1 nA (at lower brightness) using enhanced MOT loading and two-photon ionization. 16 Because of the range of applications that a Li MOTIS can address, it is useful to investigate in some detail the transport processes that limit the brightness. A full understanding of this process will not only enhance intuition for future MOT-based system design, but also allow detailed modelling for optimizing beam brightness. To this end, we study here the time dependence of the depletion and refilling of the ionization volume in the Li MOTIS described in Ref. 16. We experimentally measure the current extracted in pulsed operation and compare these measurements with a simple hard-sphere model and an analytic solution to the transport equation. Related transport dynamics have been explored in Ref. 5 with emphasis on optimizing quasi-continuous brightness, This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1116/6.0000394 considering long-distance transport in a MOT where cooling is shut off during ionization and the excitation laser beam has a significant pushing effect. Here, we concentrate on transport localized to the ionization region in a MOT that is not turned off during ionization and find that ballistic motion dominates on the length scale relevant to ionization, even in the presence of active cooling forces.

II. EXPERIMENTAL METHODS
We realize a lithium MOTIS by photoionizing neutral 7 Li atoms at the center of a 3D magneto-optical trap (MOT), as described in Ref. 16 Ions formed at the center of the MOT are accelerated in an electric field to create an ion beam, as shown in Figure 1. The MOT is located between two electrodes, one at + 3kV and the other held at ground to create a ≈ 44 kV/m electric field at the atoms (note the DC Stark shift due to this field is negligible and does not affect the behavior of the MOT). the MOT to attain steady state in between ionization sequences. In addition to observing the ion beam's time dependence, we also measure the average current with the Faraday cup positioned to capture the entire beam, using an ammeter with a time constant much longer than any switching of the ionization rate. As described below, this provides a quantitative measure of the MOT refill dynamics when a suitable pulse sequence is applied to the ionization.

III. EXPERIMENTAL RESULTS
In order to focus on cold-atom transport limitations, we have chosen to operate our Li MOTIS in a regime where such transport effects are most prominent. To reach this regime, we use tightly focused ionization lasers with high powers to create a high This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1116/6.0000394 ionization rate, and use pulsed ionization over time periods that are short compared with the MOT lifetime to avoid complications due to changes in overall MOT population. In this regime we hypothesize that the MOTIS behavior is as follows.
When the ionization volume at the center of the MOT is first exposed to a UV pulse in combination with the (continuously on) IR beam, the instantaneous extracted current is proportional to the ionization probability and the unperturbed local MOT density. Very quickly, though, ionization causes density depletion in the ionization volume and surrounding region because cold atom transport cannot replenish the density quickly enough to match the ionization rate. The current then decreases as the density depletes. After some time, a new quasi-equilibrium is reached when the reduced ionization rate matches the cold atom transport rate. When the pulse turns off, the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1116/6.0000394 depleted ionization volume begins to refill due to cold atom transport, eventually reaching the original unperturbed state of the MOT.
Using the streak camera method described above, we have measured the time dependence of the current during a two-pulse sequence to confirm this hypothesis. is plotted for the delays in (a-d). Grey bands indicate the UV laser pulse sequence for the 33 μs delay case. Note appearance of current before the laser pulse begins is an artifact of the finite ion beam size. For each delay value the two-pulse sequence is averaged over ≈ 500 repetitions with a 1.5 ms repetition period.
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1116/6.0000394 shows the results of this measurement. As expected, the pulses show a large initial transient current followed by a lower, quasi-steady-state current. Since the magnitude of the transient response provides a measure of the density within the ionization volume at the beginning of each pulse, we use the height of the transient response in the second pulse to measure the recovery of the density depletion after the first pulse ends.
Repeating the experiment for varying delay times between the first and second pulses, we Also shown in Fig. 3 are the results of the time-averaged current measurements using the Faraday cup, conducted with the same two-pulse sequence (black circles). For each value of the delay between the two pulses, we collect and average the current over 3000 repetitions of the sequence, repeating at an interval of 1.5 ms. We then determine the average current during the 60 µs that the ionization is on by correcting for the 4 % duty cycle. As with the peak ratio data, the average current has a smaller value at short delay times and then increases as the delay time is increased until it reaches an asymptotic value. This behavior is consistent with our model of the transport dynamics: the first pulse depletes the ionization volume and then the second pulse is smaller if it arrives before the volume has a chance to refill, resulting in a smaller average current. If the delay time is long enough, the volume refills to the unperturbed MOT density by the time the second pulse arrives, and the average current becomes independent of delay time.

IV. THEORY
Given the clear measurements of ionization volume refill we have obtained, it is instructive to develop a simple theoretical description of the process. We begin with an ansatz where the MOT population consists of a cloud of cold, non-interacting atoms that is essentially uniform over the region of interest around the ionization volume. We further assume that the motion of the atoms is purely ballistic, the temperature remains constant and uniform, and all magnetic restoring forces are negligible because the MOT is located close to the magnetic field zero. In reality, atoms in a MOT are embedded in an optical molasses and so should be considered as moving in a diffusive medium.
However, the diffusion constant is such that the mean free path is much larger than the size of the ionization volume, so ballistic transport is the more appropriate picture for our situation (see Appendix I). The ionization process is treated as a pure loss channel with an effective rate, and no attempt is made to model the details of the optical transitions, either in the ionization or in the MOT laser cooling.

A. Hard-sphere model
A certain amount of intuition into the time dependence of the refill dynamics in the MOT can be gained by considering a "hard-sphere" model, in which the ionization is assumed to occur entirely within a sphere of radius 0 . In this model, we can write a simple differential equation for the number of atoms in the ionization volume, ( ): where 0 = This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. correspond to a reasonable radius that is commensurate with the beam waists of the ionization lasers (6.7 µm and 9.2 µm 1/ 2 radius for the UV and IR lasers, respectively).

B. Boltzmann equation
While the hard-sphere model provides some intuition into the time dependence of the refill, its treatment of the transport oversimplifies the spatial dependence of the density and ionization probability. As a result, it does not do a good job modeling the coefficients 0 and 1 . These coefficients are actually not independent in the model, as they both depend on the basic parameters 0 , , 0 , ̅ , and . Among these parameters, only 0 and can be considered as free fitting parameters for the given experimental conditions, since 0 , ̅ , and are measured quantities. With only two free parameters, a good fit to the data in Fig. 3 is not possible, as shown by the dashed line in the figure.
To more accurately model the spatial dependence of the density and the ionization probability, we use a collisionless Boltzmann equation with a loss term to solve for the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
where is the spatial coordinate, is the velocity, represents the peak ionization probability per unit time, and is the standard deviation of the ionization profile.
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1116/6.0000394
where 0 is a normalization constant and ℎ is the thermal velocity √ . Note we use spherical coordinates in both position and velocity space to take advantage of the (assumed) spherical symmetry, and the normalization factor 0 includes the concomitant factor of 4 . To obtain the total current as a function of time, we multiply the density by the ion charge and the Gaussian ionization probability and integrate over space: After the pulse is over and the depleted region begins to refill, the distribution evolves where is the pulse duration. When the second pulse arrives, the distribution again evolves according to eq. (2) with ≠ 0, but with an initial distribution given by eq. (5)

with = Δ . Now the density as a function of time is
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. Relative density refers to density normalized to the unperturbed value. Also shown with a dashed black line is the ionization profile. (c) Spatial profiles of the relative density refill ( , ) during refill at times Δ of 0 µs, 1 µs, 2 µs, 5 µs, 10 µs, and 30 µs, relative to the end of the first pulse. Curve labels are times in microseconds. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1116/6.0000394
during the second pulse is very similar to the depletion during the first pulse, albeit starting from a lower initial level when Δ is small, and so is not shown in Fig. (4).
To calculate the average current as would be seen by the electrometer, we write and calculate the total charge in both pulses as The average current is then can be numerically evaluated with , and 0 as parameters. Allowing these three parameters to be free it is now possible to obtain a good fit of to the electrometer refill data, and the results are shown in Fig. 5.
We note that the scale factor 0 could in principle be fixed though normalization, but that would require good knowledge of the peak MOT density at the center, and also any detection inefficiencies in the ion current measurement, both of which we do not have good knowledge of. For this fit, = (0.164 ± 0.032) µs -1 and = (11.2 ± 0.8) µm (uncertainty is one standard deviation for the fit). As can be seen, the theoretical curve matches the experiment extremely well. The optimized value of is somewhat larger than the UV and IR beam radii (6.7 µm and 9.2 µm, respectively), but considering that the UV transition is highly saturated, and the actual beam overlap is more complex than a pure spherically symmetric Gaussian, the value is quite reasonable. The optimized value of the ionization rate is also quite reasonable considering the following. The total current can be estimated by noting that it should be equal to (2 )

V. SUMMARY AND CONCLUSIONS
We have presented measurements of the transport dynamics within a Li MOTIS in the form of the time dependence of the current during pulsed ionization, and the refilling of the depleted ionization volume between two pulses. We have modeled these dynamics via simple ballistic transport of cold atoms, and obtained an analytic solution that agrees extremely well with the measurements when three basic parameters are adjusted to match theory with experiment. The optimized values of these quantities are within expectations for the experiment. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1116/6.0000394
By demonstrating good agreement between a simple ballistic theory and observed behavior of the MOTIS, we have established that ballistic transport is a suitable approach toward analyzing the microscopic behavior of a Li MOTIS, understanding its limitations, and optimizing its performance. This result is important for future development and optimization of cold atom ion sources, as it establishes a pathway toward a more complete simulation framework. Having a good understating of the cold atom dynamics in these sources is crucial input for such a framework, and when it is integrated with existing ion optical models, a complete picture of source performance will emerge, enabling innovative designs.
While the present work gives support to a simple ballistic transport model, it is important to recognize that we have only seen this so far for the limited performance range of the current experimental parameters in a Li MOTIS. Magneto-optical traps have a wide dynamic range of operation, and their behavior differs significantly depending on atomic species, laser power and detuning, magnetic field gradient, lifetime, and geometry. This is particularly true if auxiliary cooling schemes are used, such as dark state cooling 20 grey molasses, 21 or cavity cooling. 22 The possibility of diffusion-limited dynamics in the presence of laser cooling cannot be ignored for all situations, and may play a significant role in systems where extremely low temperatures are sought. In some cases neither diffusion nor ballistic transport will dominate, and it will be necessary to solve a more complicated set of equations to model cold atom source behavior. In any case, our work here represents a step in the progression toward full modeling of cold atom source behavior, and will hopefully stimulate further work in this area. where is the total current (in ions per second). The result is

ACKNOWLEDGMENTS
For a solution after the first laser pulse is finished at time , we desire a solution to eq.
The distributions 1 ( , , , ), refill ( , , , , ) and 2 ( , , , , Δ , ), corresponding to the periods during the first pulse, between pulses, and during the second pulse, respectively, can be converted to spherical coordinates and integrated over velocities as was done for eq. (13) to generate the spatial distributions as a function of time in these periods. These spatial distributions can also be integrated over space to provide the total current as a function of time.  size. For each delay value the two-pulse sequence is averaged over ≈ 500 repetitions with a 1.5 ms repetition period.  Relative density refers to density normalized to the unperturbed value. Also shown with a dashed black line is the ionization profile. (c) Spatial profiles of the relative density refill ( , ) during refill at times Δ of 0 µs, 1 µs, 2 µs, 5 µs, 10 µs, and 30 µs, relative to the end of the first pulse. Curve labels are times in microseconds.
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. T r a n s i e n t P e a k R a t i o This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.