Stochastic and equilibrium pictures of the ultracold Fano-Feshbach-resonance molecular conversion rate

The ultracold molecular conversion rate occurring in an adiabatic ramp through a Fano-Feshbach resonance is studied and compared in two statistical models. One model, the so-called stochastic phase-space sampling (SPSS) [Hodby et al., Phys. Rev. Lett. 94, 120402 (2005)] evaluates the overlap of two atomic distributions in phase space by sampling atomic pairs according to a phase-space criterion. The other model, the chemical equilibrium theory (ChET) [Watabe and Nikuni, Phys. Rev. A 77, 013616 (2008)] considers atomic and molecular distributions in the limit of the chemical and thermal equilibrium. The present study applies SPSS and ChET to a prototypical system of K+K$\to$ K${}_2$ in all the symmetry combinations, namely Fermi-Fermi, Bose-Bose, and Bose-Fermi cases. To examine implications of the phase-space criterion for SPSS, the behavior of molecular conversion is analyzed using four distinct geometrical constraints. Our comparison of the results of SPSS with those of ChET shows that while they appear similar in most situations, the two models give rise to rather dissimilar behaviors when the presence of a Bose-Einstein condensate strongly affects the molecule formation.

Figure 1

Conceptual diagram representing stochastic phase-space sampling (SPSS). See steps (1)–(4) described in the text.

Figure 2

SPSS molecular conversion rates for four criteria shown altogether. The red line corresponds to the result of criterion 1, dashed green to criterion 2, blue chain to criterion 3, and two-dotted pink chain to criterion 4. From left to right the panels correspond to ${}^{40}$K-${}^{40}$K, ${}^{39}$K-${}^{39}$K, and ${}^{40}$K-${}^{39}$K, respectively. The dots in the left figure show corresponding experimental result in Ref. [6].

Figure 3

Master diagram for Q (criterion 1) showing phase-space distribution of molecule-forming pairs, $\rho (\Delta r,\Delta p)$ [Eq. (14)]. Each consists of three panels corresponding to $T>T_c$, $T\sim T_c$, and $T<T_c$. The triplet of distributions are arranged so that the left one is the Bose-Bose case, the middle one the Fermi-Fermi case, and the right one the Bose-Fermi case. The horizontal axis shows the relative spatial separation $\Delta r$, while the vertical axis shows the relative momentum $m_{\mathrm{red}}\times \Delta v$. The brighter the color, the higher the density. ${\gamma }_q=0.19$ as in Table .

Figure 4

Master diagram for L (criterion2) showing phase-space distribution of molecule-forming pairs, $\rho (\Delta r,\Delta p)$, same as Fig. 3. ${\gamma }_l=$ 0.26 as in Table .

Figure 5

Master diagram for V (criterion 3) showing phase-space distribution of molecule-forming pairs, $\rho (\Delta r,\Delta p)$, same as Fig. 3. ${\gamma }_v=$ 0.0085 as in Table .

Figure 6

Master diagram for S (criterion 4) showing phase-space distribution of molecule-forming pairs, $\rho (\Delta r,\Delta p)$, same as Fig. 3. ${\gamma }_s=$ 0.050 as in Table .

Figure 7

The upper two figures represent phase-space density distributions of individual components in the Bose-Fermi case; (a) the fermion to the left panel and (b) the boson to the right panel at $T=0.1T_c$. (c) Shows the density of the molecule-forming pairs under criterion 1. (d) Shows the density of the fermion cutaway by criterion 1 when the boson is assumed to be concentrated entirely at the origin. (c) and (d) resemble each other.

Figure 8

Pair density distributions as functions of spatial separation $\Delta r$ at $T=0.1T_c$. (From left to right) Fermi-Fermi, Bose-Bose, and Bose-Fermi. Each solid line pertains to the indicated criterion. The integrated value of the pair density distributions is normalized to the number of molecules. All the peaks representing the initial separation are significantly larger than the experimentally known final size of the molecule.

Figure 9

The left panel shows the comparison between the two assumptions using criterion 1. The red curve (FIXED) simulates the situation where ${}^{85}$Rb fails to get cooler at $T=2.2T_{c85}=1.1T_{c87}$. The dashed green curve (UNFIXED) assumes the same temperature for the two components, as in the right panel. The dots (EXPERIMENT) show corresponding experimental data in Ref. [10].

Figure 10

The left panel shows the SPSS conversion rate for the ${}^{40}$K-${}^{87}$Rb case. The right column of panels shows the evolution of the phase-space density distributions for ${}^{40}$K and ${}^{87}$Rb as functions of temperature. The upper one is for $T>T_c$, and the lower one for $T<T_c$. The rather large mass difference in ${}^{40}$K-${}^{87}$Rb results in almost nil conversion rate toward $T\to 0$ for criterion 1. The dots in the left figure show corresponding experimental data in Ref. [12]. In this figure, the lines of the SPSS molecular conversion rate are scaled by 0.70 to account for molecular losses that occur during radio frequency association; here we followed the procedure described in Refs. [11, 12].

Figure 11

Temperature dependence of the molecular conversion rate for the Fermi-Fermi case. The red line corresponds to the result of ChET, and the dashed green corresponds to the SPSS result. (a) The $\alpha =1$ result; (b) the $\alpha =$ 2/15 result. In the figure for $\alpha =$ 2/15, the characteristic temperature $T_F$ corresponds to the Fermi temperature for the majority atom.

Figure 12

The same as Fig. 11, except here for the Bose-Bose case. Upper panels correspond to $\delta =0^{-}$, lower to $\delta =0^{+}$. For the $\alpha =$ 2/15 case, $T_c$ corresponds to the critical temperature for the major bosonic component. In ChET, $T_c$ gets shifted and the plateau regions emerge.

Figure 13

Comparison between approximations and a numerical result for the ratio $r_c$. At $\alpha \simeq 1$, both initial atomic clouds have a BEC component. However, at $\alpha \simeq 0$, only the majority component has a BEC component. This causes differences in the ratio $r_c$ to appear as a function of $\alpha$.

Figure 14

Temperature dependence of molecular conversion rate for the Bose-Fermi case, the same as in Fig. 11. In the ChET, shift of the T${}_c$ and the plateau regions appear as in Fig. 12.