Description of ion motion in a Paul trap immersed in a cold atomic gas

We investigate the problem of a single ion in a radio-frequency trap immersed in an ultracold Bose gas in either a condensed or a noncondensed phase. We develop a master-equation formalism describing the sympathetic cooling, and we determine the cooling rates of ions. We show that the cold atomic reservoir modifies the stability diagram of the ion in the Paul trap, creating regions where the ion is either cooled or heated due to the energy quanta exchanged with the time-dependent potential.

Figure 1

A regularized potential (solid line) and a pure $-C_4/r^4$ (dashed line).

Figure 2

Cooling rates in one spatial direction for (a) and (b) a ${}^{138}{\mathrm{Ba}}^{+}$ ion or (c) ${}^{174}{\mathrm{Yb}}^{+}$ immersed in an ultracold gas of (a) and (b) ${}^{87}\mathrm{Rb}$ or (c) ${}^7\mathrm{Li}$ at $T=200$ nK; scattering length $a_{sc}=R^{☆}$; (a) $\Omega =2\pi \times 100$ kHz, (b) $\Omega =2\pi \times 1$ MHz, or (c) $\Omega =2\pi \times 4.2$ MHz; and (a) ${\rho }_{\mathrm{GAS}}={10}^{13}/{\mathrm{cm}}^3$ or (b) and (c) ${\rho }_{\mathrm{GAS}}={10}^{12}/{\mathrm{cm}}^3$. Points $\alpha$, $\beta$, and $\gamma$ in the top panel represent three different regions of ion dynamics (see text for details).

Figure 3

Energy of the ion (an expectation value of $H_S$, blue line), average energy with respect to the period of the radio-frequency modulation (green), asymptotic average energy (red), and an energy amplitude (pink) for parameters of points $\alpha$ (top panel), $\beta$ (middle panel), and $\gamma$ (bottom panel) marked in Fig. 2.

Figure 4

Imaginary part of the characteristic exponent for $q=0.3$ (left) without and (right) with the buffer gas (parameters as in the top panel of Fig. 2). The gray line corresponds to the stable solution for the Paul trap $\left[\text{Im}\right(\lambda )=0]$, the dark blue line corresponds to the region of the underdamped harmonic oscillator $\left[\text{Im}\right(\lambda )>0$, $\text{Re}\left(\lambda \right)\ne 0]$, the light blue line corresponds to the overdamped harmonic oscillator $\left[\text{Im}\right(\lambda )>0$, $\text{Re}\left(\lambda \right)=0]$, and the red line corresponds to the nonstable harmonic oscillator $\left[\text{Im}\right(\lambda )<0]$.