Feshbach resonances are commonly described by a single-resonance Feshbach model, and open-channel resonances are not taken into account explicitly. However, an open-channel resonance near threshold limits the range of validity of this model. Such a situation exists when the background scattering length is much larger than the range of the interatomic potential. The open-channel resonance introduces strong threshold effects not included in the single-resonance description. We derive an easy-to-use analytical model that takes into account both the Feshbach resonance and the open-channel resonance. We apply our model to
85Rb, which has a large background scattering length, and show that the agreement with coupled-channel calculations is excellent. The model can be readily applied to other atomic systems with a large background scattering length, such as
6Li and
133Cs. Our approach provides full insight into the underlying physics of the interplay between open-channel (or potential) resonances and Feshbach resonances.
Employing a short-range two-channel description, we derive an analytic model of atoms in isotropic and anisotropic harmonic traps at a Feshbach resonance. On this basis we obtain a parametrization of the energy-dependent scattering length that differs from the one previously employed. We validate the model by comparison to full numerical calculations for
6Li-
87Rb and explain quantitatively the experimental observation of a resonance shift and trap-induced molecules in exited bands. Finally, we analyze the bound state admixture and Landau-Zener transition probabilities.
We discuss
s-wave scattering in an atomic binary collision with two coupled channels, tunable by an external magnetic field, one channel open and the other closed for the incident energies considered. The analysis is performed with a stylized model of square-well potentials. This simplification allows for a pedagogically thorough discussion of the different scattering resonances that appear in coupled channels. One of the them, the potential resonances at vanishing energy, occur as a bound state of the coupled system emerges, in turned tuned at a very precise value of the external field. The other resonances, described by Feshbach theory, occur when the incident energy is near a bound state of the closed channel, as if it were decoupled from the open channel. These resonances exist for values of the external field above a particular threshold value. Besides the potential intrinsic value of this study in a quantum mechanics course, as the analysis can be performed with minor numerical calculations, it is also an aid for the understanding of current research advances in the exciting field of ultracold gases.
We give a simple description of quantum states near the dissociation threshold of deep interatomic potentials. The influence of the potential tail is separated from effects due to the interaction at short distances. We present a general formalism which is valid for weakly bound and low-energy continuum states, both for vanishing and non-vanishing angular momentum. Its applicability is demonstrated for the example of the electronic ground state of the strontium dimer (88)Sr(2). With an appropriate choice of the potential tail that includes higher-order dispersion coefficients, all short-range effects are incorporated via the threshold quantum number ν(D) and one further parameter accounting for residual short-range effects.
For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the
s-wave phase shift ${\ensuremath{\delta}}_{0}$. It exposes a universal dependence of ${\ensuremath{\delta}}_{0}(E)$ on the potential tail and the influence of effects specific to a given potential, which enter via the scattering length
a, or equivalently, the noninteger part ${\ensuremath{\Delta}}_{\mathrm{th}}$ of the threshold quantum number
nth. The formula accurately reproduces ${\ensuremath{\delta}}_{0}(E)$ from threshold up to the semiclassical regime, far beyond the validity of the effective-range expansion. We derive the tail functions occurring in the formula for ${\ensuremath{\delta}}_{0}(E)$ and demonstrate the validity of the formula for attractive potential tails proportional to
1/r6 or to
1/r4, and also for a mixed potential tail consisting of a
1/r4 term together with a non-negligible
1/r6 contribution.
The interactions between atoms and molecules with each other and with surfaces are typically deep potential wells with attractive tails behaving asymptotically as an inverse power of the distance. In such potential wells, bound state energies
En are determined by a quantization function
F(E) according to ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$, and
F(E) is dominantly determined by the singular potential tail for near-threshold states. In this paper we formulate a general theory for the contribution
Ftail(E) of the singular potential tail to the quantization function. The general expression for
Ftail(E) contains two terms: a difference of action integrals and a difference of outer reflection phases, taken at threshold and at energy
E. Close to threshold,
E=0, strongly energy dependent and nonanalytic contributions of both terms cancel, so
Ftail(E) acquires a universal form determined by a threshold length and an effective length which is related to a subthreshold effective range. For a homogeneous potential tail proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$, one universal expression for
Ftail(E) caters for all potential strengths. We give an explicit analytical expression for the important case $\ensuremath{\alpha}=6$ and present applications involving the derivation of atom-atom scattering lengths from the binding energies of high-lying bound states of the associated diatomic molecule. We also demonstrate how the dissociation energy of a diatomic molecule can be determined from spectroscopic energies of high-lying states, and we make a quantitative comparison with the performance of the LeRoy-Bernstein formula, which fails near threshold, because the strongly energy dependent and nonanalytic contribution from the action integrals is not, as it should be, compensated by terms coming from the corresponding energy dependence of the outer reflection phase.