Density-dependent response of an ultracold plasma to few-cycle radio-frequency pulses

Ultracold neutral plasmas exhibit a density-dependent resonant response to applied radio-frequency (rf) fields in the frequency range of several to hundreds of megahertz for achievable densities. We have conducted measurements where short bursts of an rf fieldwere applied to these plasmas, with pulse durations as short as two cycles. We still observed a density-dependent resonant response to these short pulses, but the time scale of the response is too short to be consistent with local heating of electrons in the plasma from collisions under a range of experimental parameters. Instead, our results are consistent with rapid energy transfer to individual electrons from electric fields resulting from an overall displacement of the electron cloud from the ions during the collective motion of the electrons. This collective motion was also observed by applying two sharp electric field pulses separated in time to the plasma. These measurements demonstrate the importance of collective motion in the energy transport in these systems.

Figure 1

A diagram of our electrode assembly. The small red rectangles represent a cross section of the copper rings that we use for our electrodes. The grey rectangles around them are the aluminum mounts, which are electrically grounded. The magnetically trapped ultracold atoms are transferred to the center of these electrodes and ionized. The electrodes and a set of wire mesh grids (blue) apply electric fields which pull electrons toward the MCP. The distance between points $A$ and $B$ in the figure is 1.5 cm.

Figure 2

A sample of the response to a delayed application of the rf field to the UCP at an applied frequency of 20 MHz ($\Delta E/k_B=100$ K, $5\times {10}^5$ total ions, 6 V/m peak-to-peak applied rf field). The solid, red curve shows the continuous application of the rf field throughout the formation of the UCP, with the resulting electron signal owing to the application of an rf field. The dashed, black curve shows the rf field being turned on at $\sim$4 $\mu$s and left on continuously afterward. Not only does the UCP have a sharp response to the initial application of the rf field, but the location of the resonant response in time can be seen to be shifted later. The sharp initial response occurs for all UCP conditions in our system.

Figure 3

A comparison of typical electron escape (red, solid) during UCP expansion to that with an applied two-cycle rf field (black, dashed). ($\Delta E/k_B=100$ K, $6.7\times {10}^5$ total ions, 5 V/m peak-to-peak applied rf field at 20 MHz). The figure is scaled to better see the electron escape after the initial prompt peak. Inset is the difference between the two signals at the time of the application of the rf field. The signal seen at $\sim$5 $\mu$s is electronic, and not from the UCP.

Figure 4

The integrated response as a function of applied rf frequency for two different times in the plasma evolution ($\Delta E/k_B=10$ K, $4.4\times {10}^5$ total ions, 5 V/m peak-to-peak applied rf field). The black circles are at 39 $\mu$s ($\delta =0.5$) in the UCP evolution and the red triangles are at 79 $\mu$s ($\delta =0.9$). These data are consistent with the proper scaling of frequency with the UCP density.

Figure 5

A plot of the ratio of the predicted resonant frequency to the frequency associated with the peak density in the UCP as given by ${\omega }_{\mathrm{peak}}=\sqrt{e^2{n}_{\mathrm{peak}}/m_e{\epsilon }_0}$. The black line shows the result of the average density calculation discussed in the Appendix below. The red $\times$'s show the results from a numerical simulation in which the electron and ion distributions are offset from each other, and a restoring force is calculated.

Figure 6

An example of the response of the plasma to two electric field pulses (FWHM of 12.5 ns per pulse) at $\Delta E/k_B=10$ K (55 $\mu$s in the plasma evolution). The integrated response here (black circles) is the response to the second pulse specifically. The red curve shows a fit of an exponentially decaying, oscillating function to the data. This fit has a frequency of $13.67\pm 0.23$ MHz as compared to a resonant frequency measurement of $13.31\pm 0.12$ MHz in this case. The decay constant of our fit in this case is $103\pm 14$ ns. The inset is an example of the set of pulses. The pulses are identical, with their peaks separated by a time $t$.

Figure 7

(a) An example of how the resonance peak time for continuous rf shifts with the applied rf voltage. At $\Delta E/k_B=10$ K (red triangles, open), one can see that the extrapolation is not linear, and the zero-power peak time is not determined by a linear fit (12 MHz applied rf field). At higher initial ionization energies (closed black circles here are at $\Delta E/k_B=400$ K) the extrapolation is linear and the zero-power peak time can be determined (16 MHz applied rf field). The inset shows the ratio of the lifetime of the UCP with an applied rf field to the lifetime without the field present (${\tau }_0=25$, 50, and 110 $\mu$s for $\Delta E/k_B=400$, 100, and 10 K respectively). The blue boxes with crosses show the lifetime reduction for $\Delta E/k_B=100$ K. In (b), we compare that extrapolation for a few different frequencies of continuous rf field (triangles, open) with the measured frequencies at specific times with the two-cycle rf method (circles, closed). These data, for $\Delta E/k_B=100$ K (red, bottom curve) and 400 K (black, top curve), show a consistent trend in time. The applied rf voltage originates from the electrode at point $A$ in Fig. 1.