Dynamics of a single trapped ion in a high-density medium: A stochastic approach

Based on the Langevin equation, a stochastic formulation is implemented to describe the dynamics of a trapped ion in a bath of ultracold atoms, including an excess of micromotion. The ion dynamics is described following a hybrid analytical-numerical approach in which the ion is treated as a classical impurity in a thermal bath. As a result, the ion energy's time evolution and distribution are derived from studying the sympathetic cooling process. Furthermore, the ion dynamics under different stochastic noise terms is also considered to gain information on the bath properties' role in the system's energy transfer processes. Finally, the results obtained from this formulation are contrasted with those obtained with a more traditional Monte Carlo approach.

Figure 1

Variation of the mean square $x$ position and $x$ velocity with respect to the $q_x=q$ parameter. The trap parameter was kept as $a=-8\times {10}^{-6}$ and $\mathrm{\Omega }=2\pi \times {10}^6$ Hz. The points are the numerical calculations, the dashed line is the first approximation described by Eq. (13), and the solid line corresponds to the second approximation given by Eq. (15). Panels (a) and (b) show $\overline{\langle v^2\rangle }$ vs $q$ and $\overline{\langle x^2\rangle }$ vs $q$, respectively. Additionally, the calculations were performed considering a ${}^{171}\mathrm{Yb}^{+}$ ion in an ultracold cloud of ${}^6\mathrm{Li}$.

Figure 2

Variation of the time-averaged mean square $x$ velocity with respect to the $q_x=q$ parameter for different stray fields. The trap parameters were fixed to $a=-8\times {10}^{-6}$ and $\mathrm{\Omega }=2\pi \times {10}^6$ Hz. The curves have been obtained numerically considering a ${}^{171}\mathrm{Yb}^{+}-{}^6\mathrm{Li}$ mixture.

Figure 3

$\overline{\langle v^2\rangle }$ vs $E$. Panel (a) shows the dependence for two different values of the $q$ parameter; the points are the numerical results while the solid lines correspond to approximate solutions of Eq. (16). Panel (b) focuses on the $q=0.25$ case; the solid lines are the fits using Eq. (16) (blue) and Eq. (17) (orange). The rest of the trap parameters are the same as those in Fig. 2 for both panels.

Figure 4

Probability density function for the ion's position and velocity after solving the Langevin equation. The fitting distributions are the approximate time-averaged distributions previously derived in Eqs. (21) and (22). Panels (a) and (b) present the $x$-position distribution without a stray field and with a stray field of 0.01 V/m, respectively. Panels (c) and (d) present the $v$ distribution for the same stray field conditions as in panels (a) and (b), respectively. For all the simulations we use the trap parameters $a=-8\times {10}^{-6}, q=0.1$, and $\mathrm{\Omega }=2\pi \times {10}^6$ Hz. A total of $1\times {10}^6$ points were used for the distributions. Again, a ${}^{171}\mathrm{Yb}^{+}-{}^6\mathrm{Li}$ mixture was considered in the calculations.

Figure 5

Expected position for the ${}^{171}\mathrm{Yb}^{+}$ ion in the ${}^6\mathrm{Li}$ cloud using white noise (blue lines) and colored noise with correlation times equal to $\tau =1.04\times {10}^{-6}$ s (red lines) and $\tau =1.54\times {10}^{-6}$ s (yellow lines). All the trap parameters are equal for each simulation: $q=0.1, a=-8\times {10}^{-6}, \mathrm{\Omega }=2\pi \times {10}^{-6}$ Hz, and $E_{\mathrm{mm}}=0.01$ V/m. The panels depict the expected position at different time intervals. Panel (a) represents the evolution from 0 to 300 $\mu \mathrm{s}$, panel (b) represents the evolution from 400 to 700 $\mu \mathrm{s}$, and panel (c) shows the behavior between 2.97 and 30.0 ms.

Figure 6

Energy relaxation for two different noise models in a ${}^6\mathrm{Li}-{}^{171}\mathrm{Yb}^{+}$ system. The agreement of both descriptions, for a given atom-ion mixture, depends of the atomic density. The ion was initially at rest in the center of the trap. However, a different initial condition will converge to the same final energy.

Figure 7

Average $x$-contribution kinetic energy as a function of time of a single ${}^{171}\mathrm{Yb}^{+}$ ion immersed in an atomic bath based on MC simulations and on solving the LE with white noise: (a) ${}^6\mathrm{Li}$ and (b) ${}^{87}\mathrm{Rb}$.

Figure 8

Kinetic energy distribution of the ${}^{171}\mathrm{Yb}^{+}$ ion in an atomic cloud of (a) ${}^6\mathrm{Li}$ and (b) ${}^{87}\mathrm{Rb}$, both at $T={10}^{-4}$ K.

Figure 9

Validity of the stochastic approach. The two parameters of the Tsallis distribution $n$ and $A$ are represented in the $y$ axis and the $x$ axis, respectively, whereas the color bar represents the average distance ($\overline{d}$) of using the stochastic approach versus the Monte Carlo method. The plotted curve represents a contour at 25% of error that we consider the limiting of the stochastic approach.

Figure 10

Colored noise resulting from solving the OU equation (B3). Panel (a) shows three different realizations of the OU noise and its distribution, and panel (b) displays the correlation function of the noise fitting with Eq. (23).

Figure 11

Radial distribution of the ion fitting by the approximate distribution (C1). To achieve an accurate fitting the trap parameters are the same as those in Fig. 4.