High-Coherence Electron and Ion Bunches From Laser-Cooled Atoms

Abstract Cold atom electron and ion sources produce electron bunches and ion beams by photoionization of laser-cooled atoms. They offer high coherence and the potential for high brightness, with applications including ultra-fast electron-diffractive imaging of dynamic processes at the nanoscale. The effective brightness of electron sources has been limited by nonlinear divergence caused by repulsive interactions between the electrons, known as the Coulomb explosion. It has been shown that electron bunches with ellipsoidal shape and uniform density distribution have linear internal Coulomb fields, such that the Coulomb explosion can be reversed using conventional optics. Our source can create bunches shaped in three dimensions and hence in principle achieve the transverse spatial coherence and brightness needed for picosecond-diffractive imaging with nanometer resolution. Here we present results showing how the shaping capability can be used to measure the spatial coherence properties of the cold electron source. We also investigate space-charge effects with ions and generate electron bunches with durations of a few hundred picoseconds. Future development of the cold atom electron and ion source will increase the bunch charge and charge density, demonstrate reversal of Coulomb explosion, and ultimately, ultra-fast coherent electron-diffractive imaging.


INTRODUCTION
The ultimate goal of X-ray and electron imaging is the ability to create "molecular movies" of the dynamics of atomic-scale processes (Dwyer et al., 2006). Molecular movies, with atomic spatial and temporal resolution, will enable dramatic advances in our understanding of critical phenomena underlying biology, materials sciences, and technological applications. For instance, rational drug design relies on knowing the molecular structure and function of membrane proteins (Pinto et al., 1992), motivating development of many different technologies including billion-dollar X-rayfree electron lasers, which attempt to produce sufficient brightness in an X-ray beam for single-shot imaging of noncrystalline objects (Chapman et al., 2011).
Electrons offer an alternative to very bright X-ray sources, which, in any case, require a bright-and lowemittance electron source. The sample interaction is 10 4 -10 6 times stronger for electrons compared to X-rays (Sciaini & Miller, 2011) but electron imaging is limited by the spacecharge effect: that is, the Coulomb interaction within an electron bunch that dramatically reduces the source brightness and coherence. Coulomb-driven explosion of the electron bunch can be reversed if the electron bunch has a uniform ellipsoidal distribution (Luiten et al., 2004).
The ability to shape electron bunches into appropriate ellipsoidal distributions is one of the motivations behind the development of a cold atom electron/ion source (CAEIS) (Claessens et al., 2005). Other advantages of a CAEIS include high source coherence due to the low temperature of the electrons and ions, and the promise of high brightness, with up to 10 6 particles/bunch. Here we present an overview of our CAEIS, investigation of space-charge effects, and the creation of ultra-fast cold electron bunches.

Cold Atom Source
In our experiments we laser-cool and trap rubidium-85 atoms. We use an effusive oven to produce hot rubidium, which is then cooled via a Zeeman slower before entering the trapping region. This provides a high-flux source of slow atoms, described in more detail in a study by Bell et al. (2010). The atoms are then confined in a magneto-optic trap (MOT) located between two accelerator plates separated by 50 mm. Using this method, up to 10 9 atoms at ∼70 μK can been trapped with a Gaussian width of <1 mm, leading to densities up to 10 11 cm −3 , similar to or greater than other CAEIS experiments (Knuffman et al., 2011;Engelen et al., 2013). Densities of 10 12 cm −3 have been achieved using a dark spot in a sodium MOT (Ketterle et al., 1993). The maximum density is important as it will ultimately limit the number of electrons or ions that can be produced for a region of a certain size. Recently, proposals have been made to use atom beams as opposed to trapped atoms to increase the flux (Kime et al., 2013;Knuffman et al., 2013).

Creating Shaped Bunches
To create electron and ion bunches, a two-stage ionization process is used (Fig. 1b). First, the trapping and cooling lasers, as well as the magnetic fields of the MOT, are turned off. A pulse of laser light resonant with the F = 3 → F′ = 4 transition (780 nm) and duration of order microseconds is then directed onto the atoms perpendicular to the accelerator plates. A 5 ns 480 nm pulsed laser beam is directed onto the excited atoms in a direction parallel to the accelerator plates. The wavelength of the pulsed blue laser can be changed over tens of nanometers to allow for either direct ionization of the already-excited atoms, or to excite them to a high-lying Rydberg state, where the static accelerator field induces field ionization.
The pulsed blue laser is focused into a ribbon of light at the MOT, with a full-width at half-maximum (FWHM) of approximately σ z = 150 μm. The size of the blue laser ribbon is important as it will determine the energy spread σ u of the electrons and ions produced via σ u ¼ eσ z F, where e is the fundamental electron charge and F the strength of the accelerating electric field. For a field of 40 kV/m, we obtain σ u = 6 meV. The temporal length of the laser pulse determines the bunch length, 10 cm and 3 mm for pulse durations of 5 ns and 150 ps.
The 780 nm excitation laser profile is transformed into an arbitrary shape using a spatial light modulator (SLM). This combination of laser wavelengths and orientations creates the shaped electron and ion bunches, as shown in Figure 1a. Approximately 10 5 electrons were produced in each bunch. The repetition rate of the experiment is 10 Hz, limited by the repetition rate of the pulsed blue laser.

Detection
We select electrons or ions by appropriate choice of polarity for our accelerator front plates (e.g. electrons in Fig. 1a). After constant acceleration, the electrons or ions are propagated for 21.5 cm in a null field, then detected on a phosphor-coupled microchannel plate detector (MCP) and imaged with a CCD camera to provide two-dimensional spatial resolution of the bunch, as shown in Figure 1c. Temporal evolution of the bunch can be determined by monitoring the potential of the grounded component of the MCP.

Temperature and Coherence Length Measurements of Cold Electron Bunches
The temperature of the electron source can be determined from the divergence of the bunches, calculated from the derivative of the edge of the image of the propagated bunch and the propagation distance. Using our ability to shape the excitation laser using a SLM, we produced a beam with sharply defined edges. The edge width before propagation was defined by the resolution of the excitation intensity profile, which in turn is defined by the optical resolution of the excitation laser imaging system, ∼10 μm. The transverse thermal velocity of the electron cloud determines the angular spread after propagation (Sheludko et al., 2010): where Q e is the detector signal proportional to charge, r the radial coordinate, e the electron charge, κ the linear magnification, d 1 and d 2 the distances through which the bunch is accelerated and freely propagated, F the accelerator field magnitude, k B the Boltzmann constant, ΔE the excess energy of the electrons after ionization, and T 0 the minimum electron temperature (i.e., when ΔE = 0). The excess electron energy can be varied by changing the wavelength of the blue laser ( Fig. 2a), and the data fitted to equation 1 to determine κ and T 0 . The minimum temperature of the electrons was found to be T 0 < 10 ± 5 K (McCulloch et al., 2011) for a bunch containing 10 5 electrons (20 fC). The electron temperature is much higher than the cold atom temperature (70 μK) owing to the intrinsic heating processes encountered during ionization, such as disorder-induced heating. From this minimum temperature we can determine the transverse coherence length of the electron bunch: where m e is the mass of the electron. Using the value for T 0 obtained above gives L c > 10 ± 3 nm. The arbitrary shaping ability of the CAEIS can also be used to directly measure the coherence length. This was achieved by using a sinusoidally shaped excitation laser and measuring the visibility of the electron pattern as a function of spatial frequency (Figs. 2b, 2c), resulting in a measurement of L c = 7.8 ± 0.9 nm following the procedure outlined in a study by Saliba et al. (2012). A coherence length of 10 nm at the source is already sufficient for imaging small biomolecules such as bacteriorhodopsin, where the unit cell length is of order 10 nm. In contrast, high-brightness conventional electron sources based on photoemission, with electron bunch temperatures of order 10 4 K, have an associated coherence length of just 0.3 nm.
Cold atom ion bunch temperatures are on the order of milliKelvins, limited by disorder-induced heating (Bannasch et al., 2013).

Ultra-Fast Cold Electrons
Ultra-fast electron diffraction enables the study of molecular structural dynamics with high resolution at sub-picosecond timescales. This is important for understanding biochemical dynamics such as protein folding and regulation, as well as the formation of cracks in novel materials (Schotte et al., 2003;Sciaini & Miller, 2011). Ultra-fast exposure times will also allow high-intensity imaging of radiation sensitive , and a fit to the recorded data (red, solid). c: Visibility of electron bunch pattern as a function of spatial frequency, with a Gaussian fit to the visibility function resulting in L c = 7.8 ± 0.9 nm. The systematic uncertainty in measuring d was 3%. From Saliba et al. (2012).
samples, such as biologically active molecules, to obtain sufficient information about the molecule before it dissociates, known as "diffract-before-destroy" imaging. To achieve this with our CAEIS, we replaced the continuous wave (CW) 780 nm excitation laser with a femtosecond laser, with a full-width-half-maximum (FWHM) of 40 nm. With femtosecond excitation, the initial electron pulse duration is limited by the spatial and temporal extent of the overlap between the new femtosecond pulses and the 5 ns pulses of 480 nm light. The overlap produces a shaped pulse of electrons or ions with a minimum duration of 150 ps (McCulloch et al., 2013). The charge of the electron bunches produced was 100 fC.
The high bandwidth inherent to short laser pulses might be expected to increase the excess energy spread of the electrons and thus destroy their transverse coherence. We performed an emittance measurement using the pepperpot method. Instead of using a physical pepperpot, we shaped the femtosecond excitation as shown in Figure 3ai and measured the spatial distribution of the electron bunches at the MCP detector. By knowing the initial and final electron beamlet distributions, the emittance ε γ can be calculated (McCulloch et al., 2013). The pepperpot measurements were performed for a series of different blue laser wavelengths, similar to the temperature measurements discussed in Materials and Methods section, and compared to results with CW excitation.
From the results (Fig. 3b) it can be seen that in region i, just below the field-free ionization threshold, the emittance increases, coinciding with an increase in ionization efficiency and therefore an increase in space-charge effects. In this region the electron bunches that are produced are ultra fast and still highly coherent. Below region i the ionisation efficiency is poor, reflected in the large error bars. In region i the blue laser couples the 5P 3/2 state to one or more fieldionizing Rydberg states, resulting in an electron bunch with minimal spread. Above threshold, in region ii, the emittance increases dramatically owing to the opening of an alternative ionization pathway: when the energy of the ionization laser is above threshold, the blue laser couples the 5P 3/2 state directly to the continuum. In this case the large near-resonant bandwidth of the 780 nm femtosecond pulse substantially increases the energy spread.
As the excess ionization energy increases further, the emittance approaches the theoretical emittance growth function: where σ is the root mean square bunch width and T the electron temperature. As can be seen, the emittance will increase with the temperature of the electron produced, which, in turn, will depend on the excess ionization energy (see Fig. 2a). The difference from this theoretical line is most likely due to space-charge effects or other heating processes that occur during ionization and extraction, which equation 3 does not take into account. This shows that the bandwidth of the femtosecond laser is not contributing appreciably to the energy spread. Below region i the emittance is approximately constant (ε r = 538 ± 26 nmrad), limited by heating during the extraction process. In the same region, the emittance with CW excitation was 141 ± 7 nmrad. Though the femtosecond emittance is larger, the corresponding coherence length is still relatively large for an electron source, at L c = 4.0 ± 0.2 nm (McCulloch et al., 2013). The difference in emittance and temperature (and therefore coherence length) between the nanosecond and picosecond bunches is because of the increased space-charge repulsion that will occur in a bunch with the same charge but density 30 times greater.

Observing Space-Charge Effects
Space-charge effects within clouds of electrons or ions cause bunch expansion. This is normally an irreversible process and leads to a loss in coherence and brightness. However, if the bunch shape is a uniform ellipsoid then the internal fields are linear, and though the bunch will still expand, the expansion can be reversed by refocusing with conventional linear-charged particle optical systems, preserving the initial coherence and brightness of the source. It has been theoretically shown that an initial bunch with a semi-circular transverse distribution and a very narrow  Figure 4. a: Gaussian bunch widths as a function of peak atom density. Blue circles indicate experimental data, with the error bars determined from the standard deviation of ∼100 measurements; dashed blue line is to guide the eye; red squares indicate general particle tracer (GPT) simulations determined from the peak density and a 0.16 ionization fraction (IF) within the interaction region determined by the sizes of the excitation and ionization lasers; green crosses indicate GPT simulations with an ionization fraction chosen to match the experimental data. The inset shows the ionization fraction chosen for each density (green points) compared with the 0.16 constant value (red dashed line). b: Measured counts from the microchannel plate detector (MCP) as a function of the simulated ion number from the ionization fraction shown in inset of (a) for an MCP potential of 1,500 (red) and 1,600 V (blue). Points indicate experimental data, error bars from standard deviation of ∼100 measurements and dashed lines indicate linear fit to data. c: Ion number, determined using the calibration from (b), as a function excitation pulse power for measured data. The right-hand axis shows the ionization fraction, determined from the atom density and ionization laser sizes. Each data point represents 100 single-shot measurements with the error bars indicating one standard deviation combined, the dashed line represents a linear fit to the data. longitudinal distribution will evolve into a uniform ellipsoid (Luiten et al., 2004). Creating such a distribution experimentally is challenging. The spatial distribution of the initial bunch depends not only on the excitation beam profile, but also on the initial density of the cold atom cloud, and the time-dependent behavior of the excitation process. We have simulated these effects using optical Bloch equations, and modeled the evolution of the bunch shape using general particle tracer (GPT) simulations (http://www.pulsar.nl/gpt).
We have investigated space-charge effects using ions rather than electrons because of their greater mass and lower velocity and consequently longer interaction times. The ion temperature is also orders of magnitude lower than for electrons, so the effects of thermal diffusion are minimal. In combination, the effects of Coulomb interactions within the bunch are much more clearly discernible.
By increasing the delay between the time when the MOT fields are turned off and the ionization beams are turned on, we are able to study the effect of atomic density on spacecharge expansion of the ion bunches by making use of the thermal expansion of the atomic cloud, this is shown in Figure 4a, which shows the bunch size for varying initial density. As expected, as the atomic density increases the bunch width also increases, in good agreement with GPT simulations for a fixed ionization fraction of 0.16 (Fig. 4a), up to a density of around 3 × 10 10 cm −3 . At higher density, we postulate that the reduced ionization fraction seen experimentally is because of absorption of some of the excitation beam by the atoms at the leading edge of the atom cloud, outside the interaction region, reducing the number of photons in the interaction region available to ionize the atoms and therefore reducing space-charge effects. The inset to Figure 4a shows the individual ionization fraction that best matched simulation and data for each initial atomic density.
By matching the simulations to the space-charge expansion data we have been able to calibrate the detection system to determine the ion number from the counts measured by the phosphor-coupled MCP and CCD imaging system. This was achieved by comparison of the integrated counts recorded on the CCD with the ion number used in GPT to obtain the correct bunch width shown in Figure 4a. The calibration is shown in Figure 4c for two different detector potentials. In both cases the R 2 coefficient was >0.99, indicating a strong linear relationship between the MCP counts and the simulated number of ions. We examined the effect between ion number and excitation power further at low power (well below the saturation limit of ∼10 mW) to illustrate how absorption of the excitation laser outside the interaction laser could lead to a reduction in ion number. As can be seen from Figure 4c, the ion number (calculated with the calibration obtained from Fig. 4b) increases linearly with excitation power. We also calculated the ionization fraction using the sizes of the ionization beams and the peak atomic density of the MOT.
Our investigations have also led to the discovery of some interesting effects such as the formation of density waves around an initially uniform circular ion bunch. This can be explained by the formation of a diffuse halo of charges around the central core of the bunch. The halo is created by reabsorption of spontaneous emission from the directly excited atoms. The dense core then expands into the halo, due to space-charge repulsion, and creates a high-density ring. We have also investigated the space-charge interaction of parallel beamlets to see the influence of overlapping self-fields. Our studies show good agreement between simulations and experiments. The simulations reveal the sensitivity of the visibility of the high-density features to the initial ion temperature: the structure is lost at temperatures of a few tens of Kelvin, highlighting the advantages of the cold atom source in comparison with conventional sources, which operate at room temperature or above, for studying these effects.

DISCUSSION
We have presented our CAEIS, including characterization of the temperature of the source and the corresponding transverse coherence lengths of the electron bunches. We have also investigated the effect of space charge on ion bunches as an analog to the much faster expansion of electron bunches, showing substantial space-charge effects. One of our main priorities is to overcome the space-charge expansion using the unique beam-shaping ability of cold atom sources to produce uniform ellipsoidal bunches. Our shaping ability is currently limited by speckle in the excitation beam image produced from the SLM owing to the hologram-production algorithm used. Overcoming this will involve implementing alternate algorithms and feedback control over the phase pattern on the SLM, by monitoring the excitation laser profile with an independent imaging detector.
Apart from space charge, another phenomenon limiting the minimum temperature of the ions produced from the CAEIS is disorder-induced heating. Nonuniform Coulomb interactions of the initially randomly distributed electrons and ions leads to an initial spread in the temperature of the bunch. For ions, this increases the temperature of the bunch by at least an order of magnitude (Bannasch et al., 2013). One way of overcoming disorder-induced heating is to use the phenomenon of Rydberg blockade, where the van der Waals' potential caused by an atom in a highly excited state prohibits nearby atoms from also being excited (Bannasch et al., 2013;Robertde Saint-Vincent et al., 2013). We have recently developed an alternative blue laser system using a frequency locking scheme based on electromagnetically induced transparency (Abel et al., 2009). With this new laser we have produced preliminary results demonstrating blockade behavior with the 30S state and measurements of the temperature effects are in progress.
By overcoming both space-charge and disorder-induced heating effects we should be able to produce ion bunches capable of sub-nanometer resolution (van der Geer et al., 2007).

CONCLUSION
We have developed a CAEIS with the ultimate goal of producing single-shot electron diffraction of biological samples.
On the path to producing these we have developed a source with a coherence length of ∼10 nm with electron temperature of 10 K. By using a femtosecond pulsed laser we have also produced ultra-fast bunches with a minimum duration of 150 ps, with a maximum coherence length of 4 nm. We have shown that space-charge effects are readily observable without the obfuscation of thermal diffusion, potentially providing a new approach to investigating subtle Coulomb interactions in high-current-charged particle sourcs. Finally, we have investigated the effects of space charge on the ion bunches produced with our system, and observed the formation of surprising structures. To improve the emittance and brightness of the source further, we are investigating reducing the temperature by using Rydberg blockade to overcome disorder-induced heating effects, and using our shaping ability to overcome space-charge effects. By implementing these advances, single-shot ultra-fast coherent-diffractive imaging with nanoscale resolution should become feasible, allowing for the creation of "molecular movies."