Ultrafast electron cooling in an expanding ultracold plasma

Plasma dynamics critically depends on density and temperature, thus well-controlled experimental realizations are essential benchmarks for theoretical models. The formation of an ultracold plasma can be triggered by ionizing a tunable number of atoms in a micrometer-sized volume of a 87Rb Bose-Einstein condensate (BEC) by a single femtosecond laser pulse. The large density combined with the low temperature of the BEC give rise to an initially strongly coupled plasma in a so far unexplored regime bridging ultracold neutral plasma and ionized nanoclusters. Here, we report on ultrafast cooling of electrons, trapped on orbital trajectories in the long-range Coulomb potential of the dense ionic core, with a cooling rate of 400 K ps−1. Furthermore, our experimental setup grants direct access to the electron temperature that relaxes from 5250 K to below 10 K in less than 500 ns.

Supplementary Material: Ultrafast Electron Cooling in an Expanding Ultracold Plasma

IONIZATION VOLUME
In order to calculate the initial electron/ion distributions ρ e/i (x, y, z) = ρ a (x, y, z) × P (x, y, z), (S1) the full 3D distribution of the atomic density ρ a (x, y, z) is modeled and multiplied with the non-linear 3D ionization probability distribution P (x, y, z) given by the intensity distribution of the laser pulse. Here, z denotes the pulse propagation direction. The ionization probabilities are obtained by solving the time-dependent Schrödinger equation. We have demonstrated in a previous work that this theoretical description is in perfect agreement with the measured ionization probabilities [1]. FIG. S1. Ionization volume. a. 2D projection of the simulated 3D electron/ion density distribution ρ e/i (x, y, z) after strong-field ionization by a single pulse with I0 = 1.9 × 10 13 W cm −2 The dashed white lines mark the cylindrical volume, which is used for the CPT plasma simulations. b. Ionization probability in x-direction (solid blue line), in z-direction (red dotted line) and normalized atomic density distribution in x-direction (dashed yellow line). Figure S1a shows the 2D-projection of the obtained electron/ion density distribution for a pulse of I 0 = 1.9 × 10 13 W cm −2 together with the cylindrical volume used for the plasma simulations. Fig. S1b depicts the ionization probability in x-direction at (y, z) = (0, 0) (solid blue line), in z-direction at (x, y) = (0, 0) (red dotted line) as well as the normalized atomic density distribution in x-direction at (y, z) = (0, 0) (dashed yellow line) as a Thomas-Fermi profile of the BEC for the experimental trap frequencies. The atomic cloud is almost spherical as the trapping frequencies are similar in all three dimensions. Whereas the photoionization can be regarded as local in in x-and y-direction, we fully ionize the atomic ensemble in the direction of the pulse propagation z.

ELECTRIC FIELD CONFIGURATION
We use the Electrostatics Module within the COM-SOL Multiphysics ® software [2] to calculate the electro-FIG. S2. Electric extraction field. Sectional view into the 3D-CAD model of the vacuum chamber (black lines). The electric potential obtained for ±Uext = 300 V is depicted as equipotential lines (from blue to red). As the ion MCP and the vacuum chamber are grounded, the electric potential is dominated by the mesh electrodes and the electron MCP.
static field configuration for each extraction voltage. For this purpose, we include the 3D computer-aided design (CAD) geometry of our setup into the simulation. The use of finite element methods (FEM) allows for the calculation of the electric potential landscape produced by the different electrodes. We used the physics-controlled mesh option 'finer' with a minimum/maximum element size of 0.88/12 mm. Figure S2 shows a sectional view into the vacuum chamber together with the equipotential lines obtained for ±U ext = 300 V. Since the high-resolution objective needs to be close to the ionization volume, the reentrant window has to be shielded by a grounded hightransparency copper mesh (identical to the ones used for the electrodes) to avoid accumulation of static charge. This leads to a non-linear extraction field strongly increasing towards the detector, as well as a top-down asymmetry, which explains the non-spherical distributions observed experimentally (compare Fig. 3).
The extraction field in the interaction region can be precisely controlled by the voltages U ext . However, its amplitude is not perfectly proportional to the applied voltage for low extraction fields (±U ext = 5 V). This is due to the voltages on the electron MCP, which give rise to an electric field on the order of 2 V m −1 in the center. In addition, the shielding by the vacuum chamber is included in the simulations.

STRAY FIELD CONTROL
The FEM simulations of the electromagnetic fields provide valuable insight into the level of control over the electric and magnetic stray fields. At an extraction voltage of ±U ext = 5 V, which corresponds to an electric field of 4.6 V m −1 at the center, the extraction field still clearly dominates over the electrical stray fields. For smaller extraction fields however, our measurements deviate from the theoretical predictions due to electric fields. In our experimental setup, these electric stray fields are passively shielded by the grounded vacuum chamber and electric gradients are controlled down to the V m −1 level.
Helmholtz coils are used to compensate magnetic fields in all three spatial dimensions with an accuracy of 10 mG. A homogeneous compensation over the extent of the detection units is enabled by meter-sized coils. We have been working here with a magnetic field offset of 370 mG along the y-axis, which increases our energy resolution and centers the electron signal onto the detectors.

CPT PLASMA SIMULATIONS
The plasma simulations are based on CPT simulations and include Coulomb interaction between the charged particles (see Methods). Besides quantities such as the mean kinetic energies of the electron/ion ensembles, the CPT plasma simulations provide detailed access to the dynamics of each charged particle. Figure S3a shows the time-evolution of the distance from the ionization center of single plasma electrons (blue lines) for the simulation depicted in Fig. 4f-g without extraction field. For clarity, the graph only depicts a random selection of 31 plasma electrons. One clearly identifies an oscillatory motion for the different particles exhibiting different frequencies ranging from hundreds of gigahertz to a few megahertz. The frequency is decreasing with increasing amplitude of the oscillations due to screening by the more closely bound electrons. The maximal ion radius (dashed red line) given by the maximum of all ion distances from the ionization center is used to distinguish between plasma and escaping electrons. In this simulation electrons are regarded as plasma electrons, if their distance at t = 2500 ns is less than twice the maximum ion radius.
In addition, the rms electron/ion radius is given for each time step (bold solid blue line / solid light red line). The asymptotic expansion velocity of the rms ion radius of v i,rms = 418 m s −1 is in reasonable agreement with the expected plasma expansion velocity v hyd = k B (T e,0 + T i,0 ) /m i ≈ 710 m s −1 for hydrodynamic expansion (dark red dotted line) at an initial electron temperature of T e,0 = 5250 K and negligible initial ion temperature T i,0 [3]. Figure  For the simulations including an extraction field, the differentiation between plasma and escaping electrons is more challenging, since electrons enter large orbits, where the non-linear extraction field become dominant, and escape from the plasma. Thus, the extraction fields reduce the number of plasma electrons by about 80% over time. For the simulations shown in Fig. 4g at ±U ext = 300 V and ±U ext = 5 V electrons are regarded as plasma electrons, if their distance from the ionization center after 40 ns / 300 ns is less than 1 mm.

EFFECTIVE SPACE CHARGE POTENTIAL
The CPT plasma simulations furthermore provide access to the space charge potential well created by the unpaired ions during the plasma expansion. The extraction field E ext adds up to the 1D space charge potential along the detection axis lowering the effective trapping potential where r denotes the distance to the ionization center in the direction of the extraction field. Figure 4g (red lines) shows the evolution of the effective space charge potential depth for the plasma simulations without extraction field as well as for extraction voltages of ±U ext = 5 V and ±U ext = 300 V (corresponding to E ext = 4.6 V m −1 and E ext = 162 V m −1 ). Here, for each time-step U (r) is approximated by the Coulomb potential of a homogeneously charged sphere where the radius R is given by the maximal ion radius and the charge Q = e·N diff is given by the difference N diff of the number of ions and electrons within the maximum ion radius. The potential depth is determined by the difference of the local maximum U eff (r max ) and the local minimum U eff (r min ) of the effective potential at

PLASMA FORMATION
For plasma formation, the depth of the space charge potential has to exceed the electronic excess energy E kin,e . Thus, for a given excess energy and a Gaussian spatial distribution of charge carriers, the creation of a minimum ion number N * = E kin,e /U 0 is required, where U 0 = 2 π e 2 4π 0σ and σ denotes the rms radius of the ionic distribution [4]. The experimentally realized ionic distribution is approximated by a Gaussian distribution with the arithmetic mean radius of σ = (2×1.35 µm+5 µm)/3 leading to a critical ion number of N * = 960. Figure S4 shows the measured brightness at ±U ext = 300 V in an elliptical area around the plasma electrons on the detector. The number of plasma electrons, as signature of the plasma formation, displays a strong dependency on the critical charge carrier density. This density has been varied either by the pulse intensity (from 0.1 × 10 13 W cm −2 to 1.7 × 10 13 W cm −2 ) or the atomic density (from 6 × 10 17 m −3 to ρ = 1.3 × 10 20 m −3 ).
In order to vary the atomic density, we modify the evaporation efficiency, which leads to a reduced number of atoms in the final optical dipole trap. The density is scanned from an ultracold thermal cloud with ρ = 6×10 17 m −3 to an almost pure condensate with ρ = 1.3 × 10 20 m −3 . The densities are determined by the analysis of the optical density distributions in a time-of-flight measurement recorded by absorption imaging for different timesteps after switching off the optical dipole trap.
Whereas the calculation of in-situ atomic densities is reliable for the limiting cases of a fully condensed or thermal atomic sample, it is known to be difficult for partly condensed samples as the determination requires a bimodal fit using a Gaussian as well as the Thomas-Fermi density model. However, the critical number of ionized atoms can be extracted from the numbers of detected electrons for each intensity density combination. As the ionization volume slightly increases with increasing peak intensity, the critical number increases as well. While for I 0 = 1.7 × 10 12 W cm −2 around 500 ions need to be created, for I 0 = 1.9 × 10 13 W cm −2 approximately 1000 electrons are required, which agrees well with the expected value of N * = 960.

PLASMA PARAMETERS
During the expansion, the densities and kinetic energies of the plasma vary over several orders of magnitude. Figure S5 displays the time evolution of central parameters extracted from the CPT plasma simulation. Figure S5a shows the evolution of the electron and ion density in the center of the plasma. The electron density drops within the initial expansion and stays constant for the first nanosecond. As the ionic component expands, the electron and ion density both decrease and even fall below typical initial UNP densities. The density evolution determines the plasma frequency as well as the plasma period given in Fig. S5b. The decrease of the electron density explains the deceleration of the electronic orbital oscillations during the plasma expansion. In addition, the ionic plasma frequency significantly decreases within the first plasma period (given by the initial ionic density). In contrast to UNP, where plasma expansion can be regarded as slow in relation to the inverse ionic plasma frequency, here, the ionic plasma period exceeds the plasma expansion duration, thus preventing ionic thermalization.
In Fig. S5c the ion and electron coupling parameters are depicted. The ionic coupling parameter decreases after the first electron plasma period when charge imbalance is established due to ionic acceleration and increasing interparticle distance. On the contrary, the electronic coupling parameter increases during the plasma expansion since the electron temperature decreases over orders of magnitude. The simulations reveal a maximum coupling parameter of Γ e = 0.3 approaching significant electron coupling.
Electron temperatures in the Kelvin domain raise the question of quantum degeneracy for the electronic ensem-ble. Fig. S5d illustrates the electron/ion de Broglie wavelengths λ dB,e/i at different expansion times. Whereas the ionic wavelength quickly decreases, the electrons reach a maximum de Broglie wavelength on the order of λ dB,e ≈ 100 nm at the end of the plasma expansion. However, the ratio of mean interparticle distance given by the Wigner-Seitz radius a WS,e/i (dashed blue/red line) and the de Broglie wavelength never exceeds 1.3 %, which yields E kin,e /E F > 6000, where E F =h

THREE-BODY RECOMBINATION
For ultracold plasma in the density and temperature regime described in this manuscript, three-body recombination (TBR) is expected to be the dominant process of electron-ion recombination. The TBR rate K TBR per ion according to classical TBR theory is given by K TBR ≈ 3.8 × 10 −9 T −9/2 e ρ e s −1 , where T e is the electron temperature in K and the electron density ρ e is given in cm −3 [3]. Figure S6 shows the calculated TBR rate per ion (solid line) as well as the time-integrated TBR probability per ion (dashed line). After 2.5 µs of plasma expansion, a cumulated TBR probability of approximately 1% is reached. As a result, the plasma lifetime is expected to be on the order of 100 µs before a significant fraction of Rydberg excitations are created. However, on the ten microsecond timescale, when the plasma is dilute and collisions barely occur, radiative and dielectronic recombination might further limit the plasma lifetime.

TIME-RESOLVED ELECTRON DETECTION
The experimental setup gives access to the distribution of arrival times of the detected electrons by a gated detection scheme (Fig. S7). For this purpose, a repulsive voltage pulse is applied to the electron extraction mesh after a variable delay t delay after the femtosecond laser pulse. The rapidly switched potential prevents electrons from passing the extraction mesh for time-of-flight durations τ ToF > t delay . The applied pulse has a duration of 2 µs and is capacitively coupled onto the meshes. This enables to measure the accumulated electron signal up to t delay , while maintaining the spatial resolution. The estimated timing uncertainty of 30 ns is caused by the temporal jitter and the pulse rise time.
In order to analyze the obtained temporal profiles (see Fig. 5b) quantitatively, a double-sigmoid function is fitted to the measured data. Figure S8 shows the fit functions for ±U ext = 100 V, ±U ext = 50 V, ±U ext = 25 V, ±U ext = 15 V, ±U ext = 10 V and ±U ext = 5 V (solid lines, from light blue to dark blue). The two inflection points for the double-sigmoid functions are given by c 1 and c 2 (vertical dashed lines). The spectra in Fig. 5c are the time derivatives of the fitted functions. The arrival time difference c 2 −c 1 determines the plasma lifetime shown in Fig. 5d. The error bars are given by the 95% confidence interval for the inflection points.

INTENSITY CALIBRATION
The experimental setup only provides access to the femtosecond laser power before passing the high resolution microscope objective. As the transmittance α T critically depends on pointing and angle of the incident laser beam, the actual peak intensities inside the vacuum chamber have to be calibrated. The averaged laser power used for the calculation of the applied peak intensity (see Methods) is given by P = α T P front . Here, P front denotes the power in front of the objective, which is measured through a circular aperture with the same diameter as the objective aperture (4 mm) at a pulse repetition rate of 100 kHz. Figure S9  well as the beam waist measured with an identical objec-tive (see Methods). The best agreement is obtained for α T = 0.1.

ULTRACOLD ELECTRON SOURCE
The electron cooling mechanisms in ultracold plasma can be exploited for plasma-based ultracold electron sources [5] producing low-emittance electron bunches. These bunches can be used to seed high-brilliance particle accelerators [6] and for coherent imaging of biological systems [7]. With the final electron temperature of T e ≈ 10 K and an rms electron bunch radius σ r = 0.52 mm, we achieve a normalized rms emittance of r = σ r · k B T e /m e c 2 = 21 nm rad and a relative transverse coherence length C ⊥ =h/(m e c r ) = 2 × 10 −5 . In our experimental setup, the electron excess energy can be reduced further by working closer to the ionization threshold, allowing to approach the value of C ⊥ = 10 −3 required for single-shot electron diffraction [8].