Range corrections to three-body observables near a Feshbach resonance

A nonrelativistic system of three identical particles will display a rich set of universal features known as Efimov physics if the scattering length $a$ is much larger than the range $l$ of the underlying two-body interaction. An appropriate effective theory facilitates the derivation of both results in the $\mid a\mid \to \infty$ limit and finite-$l∕a$ corrections to observables of interest. Here we use such an effective-theory treatment to consider the impact of corrections linear in the two-body effective range, $r_s$, on the three-boson bound-state spectrum and recombination rate for $\mid a\mid ⪢\mid r_s\mid$. We do this by first deriving results appropriate to the strict limit $\mid a\mid \to \infty$ in coordinate space. We then extend these results to finite $a$ using once-subtracted momentum-space integral equations. We take the cutoff on these equations to be large compared to $1∕\mid r_s\mid$, and find that the first-order effects of $\mid r_s\mid$ are independent of the cutoff in this regime. We also discuss the implications of our results for experiments that probe three-body recombination in Bose-Einstein condensates near a Feshbach resonance.

Figure 1

(Color online) The bound-state spectrum of the three-boson system with short-range interactions. The solid lines denote the leading-order results and the dashed lines give the next-to-leading order result for an effective range of ${\gamma }_0{r}_s=0.01$. Here the scale has been altered as per Ref. 1, with $\gamma$ and $\kappa$ first rewritten in polar form as $\gamma ={\gamma }_{0}H\cos \zeta$ and $\kappa ={\gamma }_{0}H\sin \zeta$, and then the quantities $H^{1∕4}\cos \zeta$ and $-H^{1∕4}\sin \zeta$ displayed here. The dotted line denotes the point at which trimers become unstable against breakup to an atom and a dimer.

Figure 2

(Color online) Left panel: The extracted function $G_1$ for negative scattering length. Right panel: Results for $G_1$ (solid curve) and rescaled $G_2$ (dashed curve) for positive scattering length. The vertical lines in both panels denote the thresholds at which the Efimov trimers become unstable.

Figure 3

(Color online) The recombination rate $\alpha$ as a function of $\gamma ∕{\gamma }_0$. The solid line gives the leading order result, while the dashed and dotted-dashed lines give the results for $r_s{\gamma }_0=0.01$ and 0.005, respectively.