Measurement of the Coulomb Logarithm in a Radio-Frequency Paul Trap

Samples of ultracold ${}^{174}\mathrm{Yb}^{+}$ ions, confined in a linear radio-frequency Paul trap, are heated via micromotion interruption, while their temperature, density, and therefore structural phase are monitored and simulated. The observed time evolution of the ion temperature is compared to a theoretical model for ion-ion heating allowing a direct measurement of the Coulomb logarithm in a linear Paul trap. This result permits a simple, yet accurate, analytical description of ion cloud thermodynamic properties, e.g., density, temperature, and structural phase, as well as suggests limits to and improvements for ongoing trapped-ion quantum information efforts.

Figure 1

(a) Laser fluorescence profile for a sample of ions at $T_D$ (solid line) and at $\sim 90\mathrm{K}$ (dashed line). The inset shows a typical fluorescence image of an ion cloud. (b) The observed (dots) and simulated (line) fluorescence ratios for $\delta =-30\mathrm{MHz}$ vs heating time. (c) The extracted (dots) and simulated (solid line) $T_{\sec }$, as well as (dashed line) $T_{\mathrm{tot}}$, vs heating time.

Figure 2

The experimental (black dots) and molecular dynamics (white dots) determinations of $\ln \Lambda$ versus $g$. Despite large variation in trap parameters (see text) the observed values fall along the same curve, indicating a “universal” form for $\ln \Lambda$. The red line represents the best fit described in the text, while the black and dashed line are the results of Ref. 8 and $\ln (0.765/g)$ [6], respectively. Shown at the right of the figure are the velocity distribution functions, extracted from simulation, for selected $g$ values.

Figure 3

Comparison of experimental temperature and density for two ion clouds [$N=280$ (blue dots) and $N=2800$ (red dots)] to the results predicted by Eq. (4) with $\ln \Lambda$ given by Eq. (6) (lines) as a function of time. The calculated heating ion-ion heating rate is also shown for both cases.