Effective microscopic models for sympathetic cooling of atomic gases

Thermalization of a system in the presence of a heat bath has been the subject of many theoretical investigations especially in the framework of solid-state physics. In this setting, the presence of a large bandwidth for the frequency distribution of the harmonic oscillators schematizing the heat bath is crucial, as emphasized in the Caldeira-Leggett model. By contrast, ultracold gases in atomic traps oscillate at well-defined frequencies and therefore seem to lie outside the Caldeira-Leggett paradigm. We introduce interaction Hamiltonians which allow us to adapt the model to an atomic physics framework. The intrinsic nonlinearity of these models differentiates them from the original Caldeira-Leggett model and calls for a nontrivial stability analysis to determine effective ranges for the model parameters. These models allow for molecular-dynamics simulations of mixtures of ultracold gases, which is of current relevance for optimizing sympathetic cooling in degenerate Bose-Fermi mixtures.

Figure 1

Demonstration of the transition in the dynamics from (a) a single fixed point at $(0,0)$ to (b) ones off center on changing the parameters of the velocity-independent interaction. In each of the graphs, the initial (black points) and final (red) phase-space locations of the $N_b$ bath oscillators are shown as well as the blue trajectory of a single test particle. Note that the initial condition, chosen here at random, determines the shape of the trajectory in the more complicated case (b). Both cases are evaluated for $\lambda =0.25$, one test particle, $N_b={10}^3,T_b=0.5,T_p=0.1,M=m=1,{\omega }_n=1$, and $\mathrm{\Omega }=144/89$. (a) ${\gamma }_E=2\times {10}^{-3}$; (b) ${\gamma }_E=0.1$, with temperatures, masses, angular frequencies, parameters ${\gamma }_E,\lambda$ expressed in arbitrary units.

Figure 2

Stability diagrams in parameter space for the dynamical fixed point at $(P^{*},Q^{*})=(0,0)$ for the two forms of the interaction Hamiltonians, velocity independent (a) $({\gamma }_E,\lambda$ parameter space) and velocity dependent (b) $({\gamma }_M,\lambda$ parameter space). The white background denotes values where the origin is stable (effective single-well potential). The blue region corresponds to fixed points off the origin (due to the effective double-well dynamics), while the red (chessboard-shaded) region is a boundary where our iterative scheme does not converge to within $\epsilon ={10}^{-10}$, which may be viewed as a critical boundary. Other relevant parameters in both cases shown are $N_b={10}^3,m=M=1,{\omega }_n=1,\mathrm{\Omega }=144/89$, and $T_b=0.5$. The convergence condition is set by $\epsilon$ and the maximum number of iterations is ${10}^4$. Masses, angular frequencies, temperatures, and parameters ${\gamma }_E,{\gamma }_M$, and $\lambda$ are expressed in arbitrary units.

Figure 3

Energy spectrum of initial target system (red, diamond), bath (blue, circle) and final target (black, triangle), bath (green, square) for (a),(b) velocity-independent and (c) velocity-dependent interactions. In each instance, $N_b=N_p={10}^3$ and initial temperatures set in the numerical code equal to $T_b=0.5$ for the bath, and $T_p=5$ for the target system (corresponding to inverse temperatures ${\beta }_b=T_b^{-1}=2$ and ${\beta }_p=T_p^{-1}=0.2$, respectively), $\omega =1.0,\mathrm{\Omega }=144/89$. The parameters of the interacting Hamiltonian and the running time of the simulations are (a) ${\gamma }_E=0.1,\lambda =0.01,t_{\mathrm{run}}=5\times {10}^4$, (b) ${\gamma }_E=0.1,\lambda =0.1,t_{\mathrm{run}}=2\times {10}^4$, and (c) ${\gamma }_M=0.1,\lambda =0.1,t_{\mathrm{run}}={10}^4$. The unweighted best fit to the Boltzmann distributions, shown as continuous lines around the points of each curve, provides ${\beta }_b^{\left(i\right)}=1.71\pm 0.06$ and ${\beta }_t^{\left(i\right)}=0.171\pm 0.009$ for the actual initial bath and target system inverse temperatures, respectively, and (a) ${\beta }_b^{\left(f\right)}=0.41\pm 0.03,{\beta }_p^{\left(f\right)}=0.16\pm 0.01$, (b) ${\beta }_b^{\left(f\right)}=0.48\pm 0.03,{\beta }_p^{\left(f\right)}=0.165\pm 0.007$, and (c) ${\beta }_b^{\left(f\right)}=0.049\pm 0.009,{\beta }_p^{\left(f\right)}=0.045\pm 0.003$ for the final temperatures. Masses, angular frequencies, temperatures, parameters ${\gamma }_E,{\gamma }_M,\lambda$, energies and inverse temperatures are expressed in arbitrary units.

Figure 4

Interaction energy as a function of time for velocity-independent (a) and velocity-dependent (b) interactions. The plots are for two choices of $\lambda =0.01$ (black, bottom curve) and $\lambda =0.1$ (red, top curve), with ${\gamma }_E=0.1$ (a) and ${\gamma }_M=0.02$ (b), where the latter parameter was intentionally chosen to have comparable final interaction energy in the two situations. Oscillations which appear on short time scales are shown in the insets. All other parameters are chosen as in Fig. 3, and both time (in steps) and energies are expressed in arbitrary units.

Figure 5

Relaxation dynamics for target and bath subsystems for velocity-independent interactions. The effective inverse temperature is shown vs time for the test particles with unweighted (red dashed line) and weighted (red continuous line) fitting to a Boltzmann distribution, bottom curves on the left side of each plot, and for the bath particles with unweighted (black dashed line) and weighted (back continuous line) fitting, top curves on the left side of each plot. In each instance, $N_b=N_p={10}^3,T_b=0.5,T_p=5.0,\omega =1.0,\mathrm{\Omega }=144/89$, with interaction parameters (a) ${\gamma }_E=0.1,\lambda =0.01$, (b) ${\gamma }_E=0.1,\lambda =0.1$, and (c) ${\gamma }_E=0.5,\lambda =0.1$. While the weighted fits are expected to be more accurate, significant deviations from the results of unweighted fits can be used to infer the presence of strong deviations from a Boltzmann distribution. The spread in the leftmost data points is a consequence of small statistics associated with the sampling of the thermal initial conditions through different seeds for the random number generator. The typical relative errors associated with the inverse temperature determination through the fits are about $10\%$, and masses, angular frequencies, parameters ${\gamma }_E,\lambda$, energies, and inverse temperatures are expressed in arbitrary units.

Figure 6

Relaxation dynamics for target and bath subsystems for velocity-dependent interactions. The effective inverse temperature is shown vs time for the test particles with unweighted (red dashed line) and weighted (red continuous line) fitting to a Boltzmann distribution, bottom curves on the left side of each plot, and for the bath particles with unweighted (black dashed line) and weighted (black continuous line) fitting, top curves on the left side of each plot. Like in Fig. 5, we have $N_b=N_p={10}^3,T_b=0.5,T_p=5.0,\omega =1.0,\mathrm{\Omega }=144/89$, while the interaction parameters are (a) ${\gamma }_M=0.02,\lambda =0.01$, (b) ${\gamma }_M=0.02,\lambda =0.1$, and (c) ${\gamma }_M=0.1,\lambda =0.1$. The spread in the leftmost data points is a consequence of small statistics associated with the sampling of the thermal initial conditions through different seeds for the random number generator. The typical relative errors associated with the inverse temperature determination through the fits are about $10\%$, and masses, angular frequencies, parameters ${\gamma }_M,\lambda$, energies, and inverse temperatures are expressed in arbitrary units.

Figure 7

Long-time simulation for the velocity-independent case. On the left plot (a) the effective inverse temperature is shown vs time for the test particles with unweighted (red dashed line) and weighted (red continuous line) fitting to a Boltzmann distribution, as in the two top curves on the right side of the plot, and for the bath particles with unweighted (black dashed line) and weighted (black continuous line) fitting, as in the two bottom curves on the right side of the plot. On the right plot (b) the corresponding interaction energy as a function of time is depicted. The parameters are chosen as in the case (c) of Fig. 5, and all quantities are expressed in arbitrary units.