Demonstrating universal scaling for dynamics of Yukawa one-component plasmas after an interaction quench

The Yukawa one-component plasma (OCP) model is a paradigm for describing plasmas that contain one component of interest and one or more other components that can be treated as a neutralizing, screening background. In appropriately scaled units, interactions are characterized entirely by a screening parameter, $\kappa$. As a result, systems of similar $\kappa$ show the same dynamics, regardless of the underlying parameters (e.g., density and temperature). We demonstrate this behavior using ultracold neutral plasmas (UNPs) created by photoionizing a cold ($T\le 10$ mK) gas. The ions in UNP systems are well described by the Yukawa model, with the electrons providing the screening. Creation of the plasma through photoionization can be thought of as a rapid quench of the interaction potential from $\kappa =\infty$ to a final $\kappa$ value set by the electron density and temperature. We demonstrate experimentally that the postquench dynamics are universal in $\kappa$ over a factor of 30 in density and an order of magnitude in temperature. Results are compared with molecular-dynamics simulations. We also demonstrate that features of the postquench kinetic energy evolution, such as disorder-induced heating and kinetic-energy oscillations, can be used to determine the plasma density and the electron temperature.

Figure 1

Experimental DIH curve plotted in scaled units for average one-dimensional kinetic energy per ion ${\langle \text{KE}\rangle }_{\text{fit}}$ and time $t$. Curve taken at $\kappa =0.14,n={10}^{15}{\mathrm{m}}^{-3}$, and $T_e=440\mathrm{K}$, which yields $E_c/k_B=2.7\mathrm{K}$ and $2\pi /{\omega }_{\text{pi}}=1.4\mu \mathrm{s}$.

Figure 2

$\mathrm{\Gamma }\left(\kappa \right)$ from solving Eq. (4).

Figure 3

One-dimensional velocity distribution for particles in a Yukawa OCP calculated with the MD simulation (red, solid) compared to maxwellian distribution of equivalent energy (gold, dash-dot) and the maxwellian corresponding to the best fit to the distribution (blue, dashed) for $\kappa =0.39$. Velocity is in scaled units ${\tilde{v}}_x=v/\left(a{\omega }_{pi}\right)$. (a)–(d) ${\omega }_{pi}t/2\pi =\{0.1,0.25,0.5,3\}$. Inset: The tail of the distribution, showing the relatively large populations at high velocity in the distributions calculated with MD.

Figure 4

Scaled kinetic energy ${\mathrm{\Gamma }}_{\text{gen}}^{-1}=2\langle \text{KE}\rangle /E_c$, where $\langle \text{KE}\rangle$ is the one-dimensional kinetic energy per particle and $E_c=e^2/\left(4\pi {\epsilon }_{0}a\right)$ is the Coulomb energy between nearest neighbors, vs scaled time ${\omega }_{\text{pi}}t/2\pi$. Curves at various $\kappa$ are calculated using a MD simulation that propagates the equations of motion (in scaled units) for a Yukawa OCP with random initial positions and initial velocities all set to zero.

Figure 5

Comparison of ${\mathrm{\Gamma }}_{\text{gen}}$ and ${\mathrm{\Gamma }}_{\text{fit}}$ from MD simulations to $\mathrm{\Gamma }\left(\kappa \right)$ [Eq. (4)]. ${\mathrm{\Gamma }}_{\text{gen}}=E_C/\left(2\langle \text{KE}\rangle \right)$ is the generalized Coulomb coupling parameter calculated from the averaged kinetic energy of the ions, while ${\mathrm{\Gamma }}_{\text{fit}}=E_C/\left(k_B{T}_{\text{fit}}\right)$ corresponds to the temperature extracted from fitting the velocity distribution to a Maxwellian. In the early stages of the equilibration, ${\mathrm{\Gamma }}_{\text{fit}}$ falls below ${\mathrm{\Gamma }}_{\text{gen}}$. As the system equilibrates, both measurements of $\mathrm{\Gamma }$ slowly rise to $\mathrm{\Gamma }\left(\kappa \right)$.

Figure 6

Top: $T_{\text{fit}}\left(t\right)$ for $\{n,T_e,\kappa =3\times {10}^{14}{\mathrm{m}}^{-3},105\mathrm{K},0.23\}$ and $\{9\times {10}^{15}{\mathrm{m}}^{-3},440\mathrm{K},0.20\}$. Bottom: ${\mathrm{\Gamma }}^{-1}({\omega }_{\text{pi}}t/2\pi )$ for the same parameters. The collapse of the curves in the top panel onto the nearly identical curves matching the MD simulations in the bottom panel demonstrates the universal scaling of DIH over a wide range of $n$ and $T_e$.

Figure 7

Top: $T_{\text{fit}}\left(t\right)$ for densities ranging from $n=9\times {10}^{14}$ to $9\times {10}^{15}{\mathrm{m}}^{-3}$ at $T_e=440\mathrm{K}$. Bottom: ${\mathrm{\Gamma }}_{\text{fit}}^{-1}({\omega }_{\text{pi}}t/2\pi )$ for the same parameters. The scaled data are all similar because $\kappa$ varies slowly with plasma density.

Figure 8

Top: $T_{\text{fit}}\left(t\right)$ for $n\sim 5\times {10}^{15}{\mathrm{m}}^{-3}$ at various $\kappa$. Bottom: ${\mathrm{\Gamma }}_{\text{fit}}^{-1}({\omega }_{\text{pi}}t/2\pi )$ for the same parameters. Each experimental curve matches the appropriate simulation curve.

Figure 9

Demonstration of the effectiveness of fitting DIH curves for $n$ and $T_e$. (a) and (b) Curve of best fit (blue) with curves corresponding to +10% (red) and $-10\%$ (green) variation in $n_{\text{fit}}$. (c) and (d) Curve of best fit (blue) with curves corresponding to +20% (purple) and $-20\%$ (gold) variation in $T_{e,\text{fit}}$.

Figure 10

Comparison between $n_{\text{cam}}$ and $n_{\text{fit}}$. Uncertainty in $n_{\text{cam}}$ stems from camera calibration $\left(\pm 10\%\right)$. Uncertainty in $n_{\text{fit}}$ is determined directly from the confidence intervals (95% confidence) of the fit. The gold curve shows the expected $n_{\text{fit}}$ for a plasma with additional density variation as described in the text.

Figure 11

$T_{e,\text{fit}}$ vs $T_{e,\text{dyeCal}}$ uncertainty in $T_{e,\text{dyeCal}}$ stems from uncertainty in the dye laser frequency $\left(\pm 10\text{K}\right)$. Uncertainty in $T_{e,\text{fit}}$ is determined from the confidence intervals (95% confidence) of the fit. The gold curve shows the expected $T_{e,\text{fit}}$ for a plasma with additional density variation as described in the text.

Figure 12

Comparison between the uniform density $\kappa =0.35$ MD DIH curve (blue) and the curves with regional and/or shot-to-shot density fluctuations taken into account. The shot-to-shot fluctuations are the larger source of dephasing, becoming significant when the fluctuations are on the 20% level (green).