Giant Formation Rates of Ultracold Molecules via Feshbach-Optimized Photoassociation

Ultracold molecules offer a broad variety of applications, ranging from metrology to quantum computing. However, forming “real” ultracold molecules, i.e., in deeply bound levels, is a very difficult proposition. Here, we show how photoassociation in the vicinity of a Feshbach resonance enhances molecular formation rates by several orders of magnitude. We illustrate this effect in heteronuclear systems, and find giant rate coefficients even in deeply bound levels. We also give a simple analytical expression for the photoassociation rate and discuss future applications of the Feshbach-optimized photoassociation technique.

Figure 1

FOPA: Colliding atoms (1) interact via open (blue) and closed (green) channels due to hyperfine interactions. A Feshbach resonance occurs when a bound level (2) (green wave function) coincides with the continuum state (blue wave function). A photon (wavelength $\lambda$) can associate the atoms into a bound level $v$ (3) of the ground state potential (red) with inner and outer classical turning points $R_{\mathrm{in}}$ and $R_{\mathrm{out}}$.

Figure 2

$K_{\mathrm{PA}}^v$ in ${\mathrm{cm}}^3/\mathrm{s}$ versus the $B$ field ($T=50\mu \mathrm{K}$, $I=1\mathrm{W}/{\mathrm{cm}}^2$) for various levels ($v,J=1$) of the LiNa $X{}^1\Sigma ^{+}$ potential, starting from ${}^6\mathrm{Li}(f=\frac{1}{2},m=-\frac{1}{2})$ and ${}^{23}\mathrm{Na}(f=1,m=-1)$. Two Feshbach resonances at 1081 and 1403 G enhance the PA rates by several orders of magnitude.

Figure 3

Probability density $|{\Psi }_{ϵ,l=0}(R){|}^2$ vs $B$. As $B$ nears a resonance, $|{\Psi }_{ϵ,l=0}{|}^2$ increases sharply (truncated above 0.01). Examples of $|{\Psi }_{ϵ,l=0}{|}^2$ off and on resonance (green planes at 1200 and 1400 G, respectively) are shown in Fig. 4.

Figure 4

$|{\Psi }_{ϵ,l=0}(R){|}^2$ on (black) and off (red) resonance. The top panel shows a peak appearing near $R\sim 40\mathrm{a}.\mathrm{u}.$ on resonance (inset). The upper bound level $v=44$ of $X{}^1\Sigma ^{+}$ is also depicted. The off-resonance density is negligible for $R<50\mathrm{a}.\mathrm{u}.$ The bottom panel illustrates the inner region with more deeply bound target levels (e.g., $v=0$ and 4). Again, $|{\Psi }_{ϵ,l=0}{|}^2$ is sizable on resonance and negligible off resonance.

Figure 5

In (a) scattering length $a$ for full coupled problem (circles) and the fit using $a=a_{\mathrm{bg}}(B)(1-\frac{{\Delta }_1}{B-B_1}-\frac{{\Delta }_2}{B-B_2})$ with $a_{\mathrm{bg}}(B)=a_{\mathrm{bg}}^{(0)}+a_1(B+B_0)+a_2(B+B_0{)}^2$. In (b) $K_{\mathrm{PA}}^v$ for $v=0$ at $50\mu \mathrm{K}$ and $1\mathrm{W}/{\mathrm{cm}}^2$ (circles) and the formula (8) using $\tan \delta =k{a}_{\mathrm{bg}}(B)(\frac{{\Delta }_1}{B-B_1}+\frac{{\Delta }_2}{B-B_2})$, with $C_1$ and $C_2$ $B$-independent. Numerical parameters are given in each plot.