Applying a short electric field pulse to an ultracold plasma induces an electron plasma oscillation. This manifests itself as an oscillation of the electron center of mass around the ion center of mass in the ultracold plasma. In general, the oscillation can damp due to either collisionless or collisional mechanisms, or a combination of the both. To investigate the nature of oscillation damping in ultracold plasmas, we developed a molecular dynamics model of the ultracold plasma electrons. Through this model, we found that depending on the neutrality of the ultracold plasma and the size of an applied DC electric field, there are some parameter ranges where the damping is primarily collisional and some primarily collisionless. We conducted experiments to compare the measured damping rate with theory predictions and found them to be in good agreement. Extension of our measurements to different parameter ranges should enable studies for strong-coupling influence on electron-ion collision rates.

One of the fundamental features of a plasma is the existence of electron plasma oscillations.1 These oscillations are a hallmark of collective effects and set the fundamental timescale for electron dynamics in the plasma. The plasma oscillation frequency is determined largely by the plasma density and fundamental constants for cold electron plasmas, although there is some temperature sensitivity for shorter-wavelength oscillations too.1 In addition to the oscillation frequency, plasma oscillations have a damping rate as well.2–4 This damping can be primarily collisional in nature through electron-ion collisions. For other parameters, it can be primarily collisionless in nature, most notably through Landau damping.5 In addition, there are other collisionless effects such as plasma oscillation echos6,7 and mode coupling between Tonks-Dattner modes in non-uniform density plasmas that can influence the oscillation amplitude as a function of time.8–10 The work reported here focuses on experimental and theoretical studies on the primary nature (collisional or collisionless) and rate of plasma oscillation damping in ultracold plasmas.

Ultracold neutral plasmas (UCPs) represent a useful system in which to study electron oscillations. These plasmas are created through the photoionization of either ultracold atoms11 or atoms or molecules in beams.12,13 Many plasma oscillation experimental results have been reported in UCPs, focusing on the resonant oscillation response10,14 of UCPs. The initial electron kinetic energy in UCPs can be varied over a wide range by altering the photoionization wavelength and so oscillations can be studied as the electron temperature is varied. The range of temperatures is such that at the coldest temperatures the electrons can be influenced by strong coupling physics, where electron spatial correlations become significant and typical plasma approximations begin to break down. A strong coupling parameter of up to Γ ∼ 0.2 is predicted to be achievable in the electron component of UCPs,15 where Γ is a dimensionless ratio of the nearest-neighbor Coulomb potential energy to the characteristic kinetic energy of the electrons in the plasma.

In contrast to the oscillation frequency, the damping rate of plasma oscillations in UCPs has not been studied extensively. In the experiments of Ref. 16, it was initially observed that particular electron plasma oscillations in a relatively low-density UCP parameter regime did indeed damp with time, but a systematic study of the damping rate was not conducted. As such, there were several open questions about electron plasma oscillation damping in these systems. Is the damping primarily collisional or collisionless, or does the nature of the damping depend on the parameters of the UCP? Is the damping rate temperature-dependent? How does the damping rate depend on charge neutrality? On the magnitude of an applied DC electric field?

In the work described in this article, we address these questions using a combination of theoretical and experimental techniques. Experimentally, electron oscillations were induced in a UCP by subjecting it to a short electric field pulse that imparted an impulse acceleration to the electrons. The electrons then oscillated in space with respect to the ions in the UCP. A second short electric field pulse was used to detect this oscillation through electron escape from the UCP as described in detail below. The center-of-mass motion of the electron oscillation was measured in this way and its damping rate determined.

Theoretically, a computational model was created to match the experiment conditions to study the associated physics. The computational model was used to evaluate the plasma oscillation damping dependence on electron temperature, charge neutrality, and applied electric field. The dependence on all of these factors was predicted by the model. Experimental measurements of the damping rate at different electron temperatures were then compared to model predictions. Our experimental measurements were made before we fully understood the collisionless contribution to the damping rate as revealed by our model calculations. These measurements were conducted in a parameter regime where the damping rate was dominated by collisionless mechanisms. The comparison of the model predictions and experimental measurements constitutes a test of the model, and the experimental and model damping rates were found to be in agreement. Using the model, different parameter regimes can be identified where collisional damping is predicted to be the dominant damping mechanism, and we discuss the requirements for accessing these parameters using our experimental apparatus in future measurements.

In Sec. II, we describe our computational model and its results. In Sec. III, we describe our experimental technique. In Sec. IV, we described the method by which we extract our damping rate and its result. In Sec. V, we present the analysis of our experimental results and compare them to the model predictions. In Sec. VI, we discuss our conclusions and possibilities for future work.

We developed a computational model to describe the electron motion for UCP parameters that correspond to our experimental system. In this model, the position and the velocity of each individual electron are calculated via the leapfrog method and are tracked over time.18 Electrons interact with each other via direct calculations of Coulomb interactions. To replicate experimental conditions, the electrons also interact with a DC electric field. To avoid the creation of bound states, UCP ions are modeled by a continuous charge distribution instead of as discrete ions. The ion distribution does not evolve in time, which is a reasonable approximation over the timescales of the simulation given the mass difference between UCP ions and electrons.

To match experimental conditions, the ions are assumed to have density distribution n=nier2/2σ2, where ni is the ion peak density, r is the distance to the origin, and σ characterizes the spatial extent of the ions. The influence of the ion distribution on the electrons is modeled in two ways. First, the distribution results in a position-dependent force on each electron that is included in the integration of the electrons' motion. Second, electron-ion collisions are modeled via random binary Coulomb collisions. These collisions have been incorporated into the model via a Monte Carlo collision operator which assumes stationary and infinitely massive ions. The range of these binary collisions is truncated via a cut-off parameter based on the Debye length calculated using the electron temperature and the local ion density, following typical practice to avoid divergences in Coulomb collisions.19 The size of the cutoff in our model was chosen to reproduce the collision rate in Ref. 19 and the ratio of size σ to the cutoff is greater than 10. At each time step, the probability of collision of the electron is calculated. This probability of collision is nπλ2vdt where n is the local density, λ is the Debye length (cutoff parameter), v is the electron velocity, and dt is the timestep. If a collision occurs, an impact parameter less than λ is randomly chosen and the electron velocity is deflected by amount determined by Rutherford scattering.

The division of electron-ion Coulomb interactions into a long range interaction that confines the electrons in the UCP and into binary collisions allows for probing of the underlying physics of electron-ion interactions in the UCPs. By running simulations with and without electron-ion binary collisions, it is possible to quantify the effect that binary collisions have on the collective motion of electrons in the UCP.

In performing the computation required in our model, we take the advantage of the OpenCL standard to utilize the massively parallel architecture of modern Graphic Processing Units (GPUs). GPU programming offers enormous speed increases over traditional CPU-based programming, allowing for every single electron in the experimental system to be modeled individually. As will be discussed in more detail in Section III, we deliberately used relatively small numbers of electrons and ions in our experiments, on the order of about 105 particles. For a simulation with electron numbers Ne=1.1×105, a single GPU can process 1000 time steps in about 2 min. While this could be improved by utilizing more efficient algorithms,20–23 these algorithms are more difficult to parallelize and ultimately unnecessary for the electron and ion numbers of interest. Convergence was monitored via conservation of energy and the convergence of collision damping rates.

The model is initialized as follows. Electron positions and velocities are randomly generated in a Gaussian distribution. The UCP is then allowed to evolve until the electrons come into equilibrium in velocity and space. Once equilibrated, small proportional velocity corrections are applied to each electron to obtain the desired electron temperature, if necessary. Center-of-mass oscillations are induced by applying an instantaneous velocity kick in the z direction to the electrons. We note that we compared calculations with an instantaneous velocity kick to those with an acceleration over a finite time determined by our actual electric field pulses used, and so no significant differences were found between the two for our conditions. After this kick, the position and the velocity of the center-of mass are calculated as a function time. To parameterize the calculated damping rate in the initial computation results presented in this article, we fit a decaying sinusoidal function to the center-of-mass velocity in the z direction, which is the direction in which the impulse acceleration was initially applied. We denote the damping rate extracted this way as γ.

Simulations, replicating experimental conditions, were run with a variety of different electron temperatures and applied electric fields. The fact that the ion density distribution is not uniform in space is a critical consideration in anticipating the nature of the oscillation damping in the UCP. Naively, in a uniform-density ion cloud, it would be expected that electron center-of-mass oscillations would damp as a function only of the electron-ion collision rate. This is due to the symmetry of a uniform density ion distribution and the fact that internal (i.e., electron-electron) forces cannot influence their center-of-mass. In a UCP, however, the inhomogeneity of the density distribution produces significant damping effects.

In an inhomogeneous density UCP, the restoring force on the center-of-mass of the electrons is no longer simply linear with displacement but also depends on the spatial extent and distribution of the electrons. Shape oscillations in addition to the center-of-mass one can occur. These shape oscillations in turn couple back to the center-of-mass oscillation through the changes in the electron spatial distribution. This can lead to an apparent damping of the center-of-mass oscillation as the net amplitude decreases with time, at least initially. Because the amplitude decrease is due to mode coupling, amplitude collapse and revivals were observed numerically and so the oscillation amplitude decrease was not purely dissipative. This is evident in Fig. 1 that shows the result of a calculation of the center-of-mass amplitude as a function of time for an electron oscillation in a UCP.

FIG. 1.

UCP electron center-of-mass velocity in the z direction as a function of time, in the absence of electron-ion collisions. Plasma parameters for the simulation mirrored experimental conditions for Te = 3 K with δ = 0.45. The ion numbers was 200 000, and the applied DC electric field was 7 V/m. The figure illustrates partial collapse and revival, and suggests coupling to modes.

FIG. 1.

UCP electron center-of-mass velocity in the z direction as a function of time, in the absence of electron-ion collisions. Plasma parameters for the simulation mirrored experimental conditions for Te = 3 K with δ = 0.45. The ion numbers was 200 000, and the applied DC electric field was 7 V/m. The figure illustrates partial collapse and revival, and suggests coupling to modes.

Close modal

In order to confirm the density inhomogeneity was responsible for such apparent damping, we examined the effective damping rate as a function of charge neutrality δ=(NiNe)/Ni. In the limit of NiNe, the UCP electron density is approximately uniform as the electrons are confined at the center of the UCP. As δ is decreased, the density uniformity decreases. As is evident from the results shown in Fig. 2, the effective oscillation damping rate also increases with decreasing δ.

FIG. 2.

The impact of charge imbalance δ on the effective damping rate, γ. Simulations were run with 20 000 ions, an appropriate number of electrons as defined by δ. No DC electric field was applied, and electron-ion collisions were included in the calculation. The electron temperature was 3 K for red circles and 15 K for blue squares. To extract γ, a damped sinusoid was fit to the first five oscillations of the center-of-mass motion. The figure shows the sharp increase in γ as the UCP becomes more neutral.

FIG. 2.

The impact of charge imbalance δ on the effective damping rate, γ. Simulations were run with 20 000 ions, an appropriate number of electrons as defined by δ. No DC electric field was applied, and electron-ion collisions were included in the calculation. The electron temperature was 3 K for red circles and 15 K for blue squares. To extract γ, a damped sinusoid was fit to the first five oscillations of the center-of-mass motion. The figure shows the sharp increase in γ as the UCP becomes more neutral.

Close modal

Applied electric fields magnify the influence of ion density inhomogeneity. Such fields apply non-spherically symmetric forces breaking the symmetry of the UCP. Additionally, applied fields displace the electron cloud center-of-mass equilibrium into a more non-uniform density region of the UCP. The impact of applied electric field magnitude on γ can be seen in Fig. 3.

FIG. 3.

Effective damping rate γ vs. applied DC electric field for three different electron temperatures (green square, 3 K, red circle, 6 K, and blue triangle, 15 K). In this calculation, electron-ion collisions were included. The charge imbalance δ was 0.45, and the ion number was 200 000. The damping rate increases with an increase of applied DC electric field, and decreases with increasing electron temperature.

FIG. 3.

Effective damping rate γ vs. applied DC electric field for three different electron temperatures (green square, 3 K, red circle, 6 K, and blue triangle, 15 K). In this calculation, electron-ion collisions were included. The charge imbalance δ was 0.45, and the ion number was 200 000. The damping rate increases with an increase of applied DC electric field, and decreases with increasing electron temperature.

Close modal

Figures 2 and 3 indicate that collisionless damping decreases in magnitude as a function of increasing temperature. This is likely caused by greater electron pressure at higher temperatures better preserving the shape of the electron cloud, minimizing the influence of mode coupling. Thus, our calculations so far indicate oscillation damping rate dependence on UCP charge neutrality, applied electric field, and electron temperature.

In the exploration of the model results described so far, we have not included any magnetic fields in the presented results. We did so to simplify the description of the relevant physics. In our experiments, a 9 G magnetic field in the direction of the applied electric field was introduced to better guide the escaping electrons from the plasma to the detector. Therefore, we investigated the impact that a magnetic has a on the electron center of mass oscillations. Simulations, at variety of different temperatures, were run with a uniform magnetic field oriented along the direction of electron oscillation. In a collisionally dominated regime, the addition of such a magnetic field would not be expected to have any significant impact on the electron center of mass motion given that the Larmor radius is greater than Debye screening length.24 This was checked and verified by simulation. However, in a collisionless damping dominated parameter regime, the magnetic field did have an effect. Examples of the impact of that an applied magnetic field has on the electron oscillation can be seen in Fig. 4. While a magnetic field does not add any qualitatively new physics, it shows that the addition of a magnetic field results in an observable change in the electron oscillation. The reduction is consistent with the magnetic field, reducing the extent of shape oscillations.

FIG. 4.

UCP electron center-of-mass velocities in the axial direction as a function of time from two different simulations. The simulation at the top incorporated a 9 G uniform magnetic field pointing in the axial direction, while the bottom simulation did not include any magnetic field. Otherwise, both simulations were identical, with Te = 2.86 K, δ = 0.55, and 200 000 ions.

FIG. 4.

UCP electron center-of-mass velocities in the axial direction as a function of time from two different simulations. The simulation at the top incorporated a 9 G uniform magnetic field pointing in the axial direction, while the bottom simulation did not include any magnetic field. Otherwise, both simulations were identical, with Te = 2.86 K, δ = 0.55, and 200 000 ions.

Close modal

To create the ultracold plasma, 85Rb atoms were first cooled in a magneto-optic trap.25 These cooled atoms were then loaded into a magnetic trap that was in turn translated to a different region of the vacuum system.26 There, the atoms were photoionized using two-color photoionization involving a diode laser tuned to the 5S1∕2–5P3∕2 transition and a pulsed dye laser that ionized the atoms from the 5P3∕2 state. Details of this system are described in Refs. 16 and 27. Through tuning the wavelength of the pulsed dye laser, the initial kinetic energy of the electrons was controlled to control the UCP electron temperature. A DC electric field with an average value of ∼4.1 V/m was applied to the plasma.17 Due to the electrode used to produce this field, there was spatial variation across the UCP such that the axial electric field component varied from 0 V/m to 9 V/m. Our model indicated that this variation did not affect the damping rate qualitatively, and the variation was included in all of our model calculations that were compared to the experiment. Both this DC electric field and the 9 G magnetic field guided electrons that escaped from the UCP to a microchannel plate (MCP) detector. The signal from this detector (proportional to the electron escape rate from the UCP) formed the basis for all of our collected data. The total number of ions and electrons in the UCP and its charge balance as a function of time were determined from the integrated MCP signal as in Ref. 16.

To produce and read out electron oscillations, we applied two short electric pulses (8 ns FWHM) to the UCP starting 3 μs after formation. The time between the two pulses was varied deliberately as part of our measurement technique. The first pulse had a field of 5 V/m and initiated the electron center-of-mass oscillation. The second pulse modified the oscillation. Center-of-mass motion of the electrons produces internal electric fields that drive electrons out of the UCP. If the second pulse was timed such that it produced an acceleration in phase with the electron motion, then the oscillation amplitude would increase and the number of electrons that escaped from the plasma would thus also increase. If the second pulse is timed so that it was applied out of the phase to the electron motion, then the oscillation amplitude would decrease, and so less electrons would escape. The number of escaped electrons that resulted from the application of the second pulse was thus an indication of the amplitude of center-of-mass motion.

In using our two-pulse technique, we assume that the electron escape signal generated by the two pulses scales linearly with the amplitude of the center-of-mass motion of the electrons. We checked this assumption in three different ways that each confirmed a linear scaling. First, we performed numerical modeling of the plasma, using the model described above, in response to a two-pulse sequence. We then compared the variation of the escape and the amplitude of electron center-of-mass motion and found a linear scaling. Our second test involved measuring the electron escape response as a function of the amplitude of an applied electric field two-cycle RF pulse.16 The RF pulse does not accelerate loosely bound electrons out of the UCP as easily as a single pulse and so the total electron escape from the RF pulse is more akin to the late-time (i.e., post-initial oscillation) data from the two-pulse measurement. We found that the electron escape signal varied linearly with the applied field, again indicating a linear response. Finally, in the model, we followed the individual electron that escaped due to the oscillation, and found that the fields accelerated them were consist with linear scaling of escaping rate and the amplitude of center-of-mass motion.

We performed our measurements at average initial electron kinetic energy of 3.26 K ⋅ kB, 8.26 K ⋅ kB, and 23.26 K ⋅ kB, where kB is a Boltzmann constant. The average electron temperature is related to the average electron kinetic energy through ΔE = 1.5kBTe. The experimental data obtained in this way are shown in Section IV. From such data, we can extract both the damping rate and the oscillation frequency. Since the center-of-mass oscillation frequency is density dependent (as shown in Fig. 5), we can calculate the average density from the frequency. The charge imbalance and the total ion number are known so we could measure the size of the plasma σ at the time of the two pulse measurement as well.16 Simulations showed that for our conditions, a correction was needed in determining the density as compared to the method in Ref. 16, and so we applied this correction in our data analysis.

FIG. 5.

The center-of-mass oscillation frequency varies with plasma density. The average density for the blue line was 1.8 × 1013 m−3 and the red line was 10% denser than the blue one. The electron temperature was Te = 2.86 K in this figure.

FIG. 5.

The center-of-mass oscillation frequency varies with plasma density. The average density for the blue line was 1.8 × 1013 m−3 and the red line was 10% denser than the blue one. The electron temperature was Te = 2.86 K in this figure.

Close modal

To interpret the data, we need to estimate the plasma electron temperature. The temperature was expected to be the net result of several heating and cooling mechanisms such that we can write the electron temperature Te as a sum of several contributions.

Te=ΔTionize+ΔTdih+ΔTcont+ΔT3bdΔTadΔTevp,
(1)

where Te is the electron temperature.

The first term ΔTionize is determined by the excess energy from the ionization photon above the ionization threshold. In order to calibrate the ionization laser wavelength with respect to the ionization threshold, we measured number of ionized atoms at very low (sub-plasma) densities with respect to the laser wavelength. For these low-density conditions, far below threshold there were no ionized electrons, where above threshold there was a constant number of ionized electrons Nth. We took the data as a function of applied electric field and extrapolated to zero field. The ionization threshold was associated with the wavelength that produced Nth/2 electrons. We estimated the precision of our calibration to correspond to kB ⋅ 0.2 K, with most of the uncertainty due to the electric field extrapolation.

The second term ΔTdih is determined by disorder induced heating.28 Disorder induced heating comes from the conversion of the correlation energy of the initially disordered electrons to their kinetic energy as they reach a more ordered state. During plasma formation, the electrons are not fully in thermal equilibrium. Therefore, we used a dynamic screening length from Ref. 29 for approximation. By using tabulated value from Ref. 30, we estimate about 0.02 K of disorder induced heating for our conditions.

To estimate of what to expect for the contribution of heating from continuum lowering31 ΔTcont, a molecular dynamic simulation was needed since the electrons were not created in thermal equilibrium. We thus took the molecular dynamic simulation calculated continuum lowering from Ref. 32 and scaled it to our experimental conditions. Doing so, we found 0.8 K heating contribution from continuum lowering.

Heating from three-body recombination ΔT3bd is expected in our UCPs. Using the three-body recombination rate from the literature33 and assuming each event results in energy increase by the bottleneck energy 3kBT,34 we can estimate the amount of electron temperature increase from three body recombination. This heating was only found significant in the lowest temperature data that were collected. For those data, three body recombination was predicted to lead to a 6.1% increase in the electron temperature.

The adiabatic cooling ΔTad due to the expansion of UCP was ignorable, because we took the measurement soon after formation, and so the size did not have time to change significantly. We confirmed this was the case by calculating the electron temperature decrease following the treatment in Ref. 15.

For low density UCPs, evaporation can have a significant effect on electron temperature.17 However, for the particular experimental conditions in this work, the effect of evaporative cooling was greatly reduced due to the magnitude of the applied electric field. Using the observed escaping rate of electrons from UCP at the time that our oscillation damping data were taken, we could estimate the amount of energy carried away by the escaping electrons to produce a ΔTevp in a self-consistent model that related UCP electron potential depth, electron temperature, and escape rate to the observed conditions. We found that evaporation reduced the electron temperature by 9% in each of our experimental conditions. The temperature reduction was not sensitive to the precise value of the predicted electron escape rate.

We collected our data using three different photoionization wavelength settings and thus three different temperatures, which were 2.87 K ± 0.25 K, 5.74 K ± 0.32 K, and 14.8 K ± 0.73 K, respectively, as determined following the analysis presented above. In other words, these temperatures are determined via knowledge of the UCP density, photoionization laser wavelength, three-body recombination heating, and electron evaporation rate as measured in our experiments. The ion number was centered around 200 000 for all cases, and the data set to data set variation in average number was on the order a few percent. The charge imbalance δ was 0.475, and again was maintained on average at the percent level across data sets. The value of σ that characterizes the spatial size of the UCP was 720 μm for our data, with variation described in Section IV below.

In Fig. 6, we show all of the complete data sets that we collected for this measurement. In each, the experimental data are represented as points with error bars that correspond to the observed statistical variation averaged across all points. The best-fit model predictions corresponding to each experimental condition are also shown for comparison. Overall, there is qualitative agreement between the experimental measurements and the prediction oscillation damping. In the rest of this section, we discuss how we obtained a more quantitative comparison between the two.

FIG. 6.

Comparison of model (solid line) to the experimental data (solid points). The electron temperature of raw A, B, and C were 2.87 K, 5.74 K, and 14.8 K, respectively. Data were taken with δ = 0.475, 200 000 ions, 9 G magnetic field, and electric field 4.1 V/m.

FIG. 6.

Comparison of model (solid line) to the experimental data (solid points). The electron temperature of raw A, B, and C were 2.87 K, 5.74 K, and 14.8 K, respectively. Data were taken with δ = 0.475, 200 000 ions, 9 G magnetic field, and electric field 4.1 V/m.

Close modal

Before doing so, we would note that the signal-to-noise measurement for each data set is such that about 9 h of continuous collection is required for each set. During that time, depending on lab conditions, there could be significant drifts and so data collected during those times would not be of sufficient quality for a complete set. Those data are not shown in Fig. 6.

In order to compare the measured damping rate to the damping rate predicted by our model calculations, it is useful to parameterize the measured damping rate in terms of an effective damping constant keff. Unlike the zero magnetic field case, the center-of-mass oscillation damping is not well described by a decaying sinusoid with a single decay constant γ when the magnetic field is present (see Fig. 4). In fact, the center-of-mass amplitude change in time with the magnetic field present is generally complicated. However, the range of our experimental data corresponded to early time parts of the center-of-mass motion, and for the needed ranges of time the center-of-mass motion with the magnetic field present could be parameterized with a two-time-constant function given by

(Aek1(ω)t+Bek2(ω)t)cos(ωt+ϕ),
(2)

where A, B are the amplitude constants, k1(ω),k2(ω) are the damping constants, and ω,ϕ are the oscillation frequency and phase. The damping constants are mild functions of the density. Since ω tracks the density, the values of the damping constants and ω are linked in the model calculation. Hence, the indication of the scaling of the damping constants with ω is shown. The slight deviations of the model center-of-mass damping curves from Equation (2) are small enough not to be significant in this analysis for our current precision.

Given Equation (2), we use a two-step process for determining the effective decay constant keff from our measured data. First, we fit our measured oscillation signal (Fig. 6) to the following functional form:

D(Aek1(ω)t+Bek2(ω)t)ekauxtcos(ωt+ϕ),
(3)

where D, ϕ, ω, and kaux are all treated as fit parameters where the other parameters are determined from model predictions at the temperature associated with the experimental data being fit. D is a scaling parameter for our MCP signal. ω and ϕ allow for determination of the density and time offset associated with the data being fit. In the event that the model damping rate was a perfect fit to the experimental damping rate, kaux would be zero. In the event that the experimental damping rate fits to a rate faster than predicted, kaux would be greater than zero. If the experiment damping rate fits to a slower damping rate, then kaux would be less than zero. From all of the data sets measured at a particular temperature, the central value of kaux that best fits the entirety of the data at those conditions and the standard uncertainty in that parameter were determined. To determine keff determined by the data, we use the following equation:

(A+B)ekefft0=(Aek1(ω)t0+Bek2(ω)t0)ekauxt0,
(4)

where t0 is set to a convenient value, 150 ns for the range of times typical of our experiments. Recall that A and B are not fit parameters, but come from model predictions. The value of keff determined from the data can be compared to model predictions by deriving a model prediction for keff determined by setting kaux to zero in the equation above.

One other consideration is added to the parameterization described above. The spatial size of the atom cloud that is ionized (σ) is not perfectly constant from experiment run to experiment run. This leads to a frequency variation from run to run that appears as damping since later time data points from runs with different frequency will have a different phase from one another. We determine the damping from this effect via an additional fit parameter added to our data analysis. The ∼5% run-to-run variation indicated by this fit parameter is consistent with the estimated σ variation in our system.

This somewhat involved data analysis technique allows us to determine an effective experimental and model decay rate keff in analogy to a single sinusoidal decay constant γ for each of our temperature conditions using all of our complete experimental data sets, in light of the structure of the predicted center-of-mass damping curves from our model predictions. The model predictions and experimental data can thus be compared in terms of a damping rate.

Fig. 7 shows the results of the comparison of our measured damping rates to predicted damping rates in terms of the keff parameter described above. Within the uncertainty of our measurements, the experimental results and model predictions are in agreement. Choosing a different value of t0 does not change the degree of agreement in any significant way. This indicates that our modeling of the center-of-mass motion is in agreement with experimental results at our level of statistical significance.

FIG. 7.

Effective damping rate vs temperature. Data were taken with δ = 0.475, 200 000 ions, 9 G magnetic field, and electric field 4.1 V/m. The black dots are the experimental results and the triangles are the calculation results.

FIG. 7.

Effective damping rate vs temperature. Data were taken with δ = 0.475, 200 000 ions, 9 G magnetic field, and electric field 4.1 V/m. The black dots are the experimental results and the triangles are the calculation results.

Close modal

The conditions that we selected to take these data comparing our model to experiment were based on considerations of charge neutrality, applied electric field, and signal-to-noise. For the given number of ions and electrons we selected, significantly smaller electric fields would have resulted in more neutral plasmas and faster damping rates (see Figure 2), reducing signal-to-noise. Significantly larger electric fields would have driven faster damping due to the dependence of the damping rate on applied electric field. We operated between those extremes. Running with smaller electron and ion numbers than our system was capable of (∼10% of our maximum number of ions and electrons) resulted in being able to access a more favorable neutrality/electric field parameter region.

For this set of conditions, the damping rate is dominated by collisionless mechanisms, according to model calculations. This can be determined by comparing keff rates with the binary electron-ion collisions included in the model and without those collisions included. Removing the electron-ion collisions in the calculation reduced keff only by several percent, indicating for these conditions electron-ion collisions are predicted to have only a mild effect on the damping rate.

While our data serve as an experimental test of our model predictions, measuring the electron oscillation damping rate in a collisionally dominated parameter region would enable the testing of electron-ion collision rate predictions.19 In order to gain sensitivity to electron-ion collisions, a different UCP strategy would need to be used. The key consideration would be to run with sufficiently non-neutral plasmas but with low applied electric fields. This could be accomplished in part by running with even lower numbers of ions and electrons, but there are practical limits to how far that could be reduced and maintain a sufficient signal size in our apparatus. Instead, time-dependent applied electric fields could be used. The UCP could be created with a sufficiently high DC electric field to reduce the neutrality, and that DC field could then be slowly lowered before the damping measurements were conducted. This combines both low electric field and low neutrality conditions. In the absence of an experimental test of model predictions by experiments, however, it was unclear whether pursuing such a more complicated time-dependent applied electric field pattern would have been worthwhile. The work reported here indicates that such studies are reasonable to conduct. Preliminary investigations indicate that using such a strategy extends the range of parameters to regions where there is sensitivity to electron-ion collisions in the damping rate.

In conclusion, we investigated the damping of electron center-of-mass oscillation both theoretically and experimentally. With the aid of GPU processors, we were able to simulate our UCP system with the same electron number that we used in the experiments in a reasonable amount of computing time. Our calculations indicate that electron oscillation damping can occur through both collisional and collisionless mechanisms, with the relative importance of the two determined by UCP charge neutrality, electron temperature, and applied DC electric field.

Experimentally, we used pairs of short electric field pulses to first initiate and then detect the electron plasma center-of-mass oscillations to measure their damping rate. We found agreement between simulations and experiments. Our computational model showed that the damping observed in current experiments was dominated by collisionless mechanisms. Improvement in the range of accessible parameter space should be achievable with a more sophisticated time-dependent electric field applied to the UCP, allowing for electron-ion collision rates to be experimentally measured in this system.

There are multiple motivations for doing so. The temperature dependence of the damping rate increases in such a case, meaning that this technique could be used to measure electron temperature. The electron-ion collision rate could be determined and compared to theory predictions. By carefully choosing the plasma number, charge imbalance, and applied electric field, conditions can be selected such that strong-coupling corrections to the conventional Landau-Spitzer collision rate can be tested.1 Finally, through measurements of the electron temperature, the predicted contributions to the initial UCP can be tested experimentally, providing measurements of the continuum lowering heating in UCPs.

This work is funded by Airforce Office of Science Research, Grant No. FA 9550-12-1-0222. We also want to thank Dominic Meiser of Tech-X for the technical support and computation theory in setting up our numeric model.

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