Kinetic model of trapped finite-temperature binary condensates

We construct a nonequilibrium theory for the dynamics of two interacting finite-temperature atomic Bose-Einstein condensates and use it to numerically estimate the relative rates of the arising collisional processes near equilbrium. The condensates are described by dissipative Gross-Pitaevskii equations, coupled to quantum Boltzmann equations for the thermal atoms. The density-density interactions between atoms in different components facilitate a number of transport processes of relevance to sympathetic cooling: in particular, considering realistic miscible and immiscible trapped atomic ${}^{87}\mathrm{Rb}\text{−}{}^{41}\mathrm{K}$ and ${}^{87}\mathrm{Rb}\text{−}{}^{85}\mathrm{Rb}$ condensate mixtures, we highlight the dominance of an intercomponent scattering process associated with collisional “exchange” of condensed and thermal atoms between the components close to equilibrium.

Figure 1

Schematic representation of the collisional model. The two trapped condensates are denoted $\left(a\right)$ (left) and $\left(b\right)$ (right). Arrows indicate the various mean-field and collisional transport processes occurring within and between condensates and noncondensates of different components.

Figure 2

Each diagram represents a kinetic energy and momentum conserving collision between atoms. Squares and circles represent condensate and thermal atoms, respectively. Component $a$ particles are solid blue, while $b$ particles are open red. Diagrams for the $b$ component are obtained by interchanging the two colors in each square and circle.

Figure 3

Miscible (left) and immiscible (right) ${}^{87}\mathrm{Rb}{\text{−}}^{41}\mathrm{K}$ mixture in an isotropic harmonic trap (trap frequency $\omega =2\pi \times 20$ Hz) at temperature $21\mathrm{nK}$ with scattering lengths $a_{\mathrm{Rb}87}=99a_0,a_{\mathrm{K}}=60a_0$, and $a_{\mathrm{Rb}\text{−}\mathrm{K}}=20a_0$(miscible) or $163a_0$(immiscible) [13, 14]; each species has a total of $N={10}^5$ atoms. While our model does not include critical fluctuations required for an accurate determination of the critical temperature, an estimate for this can be obtained from numerical fits of our self-consistent condensate fractions by $1-{(T/T_c)}^{\alpha }$, where $T_c$ and $\alpha$ are fitting parameters; we find $T_c\approx 39$ nK, which is lower than the mean-field single-component $T_c$ [52] by at most 5%. Top: Condensate and thermal densities. Middle: Spatially resolved collision rates between condensate and thermal atoms. Bottom: Spatially resolved collision rates between thermal atoms.

Figure 4

(a) Comparison of integrated collision rates among the various processes at different temperatures and miscibilities for (left) ${}^{87}\mathrm{Rb}{\text{−}}^{41}\mathrm{K} (a_{\mathrm{Rb}}=99a_0,a_{\mathrm{K}}=60a_0)$ and (right) ${}^{87}\mathrm{Rb}{\text{−}}^{85}\mathrm{Rb} (a_{\mathrm{Rb}87\text{−}\mathrm{Rb}87}=99a_0,a_{\mathrm{Rb}87\text{−}\mathrm{Rb}85}=213a_0$ [11]); other parameters are as in Fig. 3. Bottom: Corresponding temperature dependence for ${}^{87}\mathrm{Rb}{\text{−}}^{41}\mathrm{K}$ mixtures of (b) ratio of integrated collision rates of ${\mathbb{C}}_{12}^{\mathrm{RbK}}$ to $C_{12}^{\mathrm{RbK}}$ and (c) collision times ${\left(\omega \tau \right)}^{-1}$ of $C_{22}^{\mathrm{RbK}}$(dots), $C_{12}^{\mathrm{RbK}}$ and $C_{12}^{\mathrm{KRb}}$ (dashes and dashed-dots), and ${\mathbb{C}}_{12}^{\mathrm{RbK}}$ (solid) collisions.