We present a two-channel model to describe the quantum state of two atoms with finite-range interaction near a Feshbach resonance. This model provides a simple picture to analytically derive the wave function and the binding energy of the molecular bound state. The results agree excellently with the measurements and multichannel calculations. For small binding energies, the system enters a threshold regime in which the Feshbach molecules are identical to long range atom pairs in single channel. According to their threshold behavior, we find Feshbach resonances can be classified into two types.
We present a two-channel model to describe the quantum state of two atoms with finite-range interaction near a Feshbach resonance. This model provides a simple picture to analytically derive the wave function and the binding energy of the molecular bound state. The results agree excellently with the measurements and multichannel calculations. For small binding energies, the system enters a threshold regime in which the Feshbach molecules are identical to long range atom pairs in single channel. According to their threshold behavior, we find Feshbach resonances can be classified into two types.
Using a two-channel model, we investigate theoretically the binding energy of confinement-induced Feshbach molecules in two- and one-dimensional ultracold atomic systems near a Feshbach resonance. We show that the two-channel prediction will evidently deviate from the simple single-channel theory as the width of Feshbach resonances decreases. For the one-dimensional system, we perform a full two-channel calculation, with the inclusion of bare interatomic interactions in the open channel. Away from the resonance, we find a sizable correction to the binding energy, if we neglect the bare interatomic interactions as in the previous work [D. B. M. Dickerscheid and H. T. C. Stoof, Phys. Rev. A 72, 053625 (2005)]. We compare our theoretical results with the existing experimental data and present predictions for narrow Feshbach resonances that could be tested in future experiments.
The susceptibility of ultracold gases to precise manipulation via external fields makes them ideal systems to study fundamental quantum mechanical phenomena. Various regimes, which are not possible to reach in condensed matter or nuclear physics, can be probed with these quantum gases. Especially the tunability of the interactions and confining potentials make that these ultracold gases offer unique possibilities to explore and study universal behavior. The tunability of the dominant two-body interactions in dilute ultracold atomic gases stems from the presence of Feshbach resonances. Via these resonances pairs of ultracold atoms can be associated to form ultracold molecules. In fact, measurements of Feshbach resonance positions can be used to reveal part of the underlying molecular structure. Vica versa, accurate molecular Born-Oppenheimer potentials can be used as input for full numerical coupled-channels calculations to reliably predict Feshbach resonances. For most of the heteronuclear mixtures currently studied however, the relevant molecular potentials are not accurate enough. As the coupled-channels method is often too elaborate and too time consuming to be used to fit the measured data to enable the construction of more accurate potentials, a need for accurate models which complement this established method is created. In this thesis we have developed a simplified model for resonant scattering of two ultracold atoms which fulfills this need. The asymptotic bound-state model (ABM) is a hierarchical model which can be systematically increased in complexity and accuracy. In its most simple form, the ABM only needs two parameters to predict the position and strength of Feshbach resonances: Two binding energies corresponding to the least bound state of the two molecular potentials involved. We demonstrate that this model, although it is solely based on bound states, can incorporate threshold effects. Additionally, the model can be extended to include: Multiple vibrational levels, overlapping resonances, and magnetic dipole-dipole interactions. This versatile model allows for a description of Feshbach resonances in a large variety of systems without accurate knowledge of the relevant molecular Born-Oppenheimer potentials. Being computationally light, the ABM can be used to analyze measurements of Feshbach resonances and hereby expose the basic underlying molecular structure. Additionally, the ABM is well suited to map out all available Feshbach resonances in a system. This ability is particularly useful when an optimal resonance is required for a certain experiment. For the 6Li-40K mixture we have utilized this ability to experimentally observe a strong Feshbach resonance which offers promising perspectives to reach the strongly interacting universal regime in a mass-imbalanced Fermi gas. By analyzing the various inelastic decay processes and the accuracy of the ab initio Born-Oppenheimer potentials, we predict a strong resonance in a mixture of metastable bosonic and fermionic helium atoms. For the homonuclear (bosonic as well as fermionic) helium gas we do not find strong resonances. For an ultracold gas of 40K atoms, an elaborate combined theoretical and experimental effort is presented to map out the Feshbach resonance structure. On the experimental side we observed 36 Feshbach resonances, 31 of which where observed for the first time. These observations are in excellent agreement with coupled-channels calculations which use unmodified Born-Oppenheimer potentials as input. On the theoretical side, we present a comparison between different simplified models of resonance scattering and the full numerical coupled-channels calculations. We have developed the resonant state model (RSM) to describe the resonant scattering of two ultracold atoms. This model is founded on resonant states, the natural (eigen) frequencies of the two-body system, and their non-trivial properties. Threshold effects are implicitly build into the RSM. The RSM can describe the effect of the continuum of scattering states, states which are neglected in the ABM, on the position and strength of a Feshbach resonance by only a few resonant states. Especially when a low-energy resonance is embedded in the continuum of scattering states, the effect of these states on the Feshbach resonance cannot be neglected. The RSM can be viewed as the natural generalization of the ABM. As the RSM can be based on just a few resonant states, it retains the computational simplicity of the ABM. Using the tunability of two-body interactions induced by a Feshbach resonance, we describe a crossover for a rotating Fermi gas between two strongly-correlated many-body states. We demonstrate that the correlation induced long-range interaction will shift the crossover with respect to the non-rotating case.
The physics of Feshbach resonance is analyzed using an analytic expression for the s-wave scattering phase shift and the scattering length a which we derive within a two-channel tight-binding model. Employing a unified treatment of bound states and resonances in terms of the Jost function, it is shown that, for strong interchannel coupling, Feshbach resonance can occur even when the closed channel does not have a bound state. This may extend the range of ultracold atomic systems that can be manipulated by Feshbach resonance. The dependence of the sign of a on the coupling strength in the unitary limit is elucidated. As a by-product, analytic expressions are derived for the background scattering length, the external magnetic field at which resonance occurs, and the energy shift ε-ε(B), where ε is the scattering energy and ε(B) is the bound-state energy in the closed channel (when there is one).
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.
It is well-known that cold atoms near s-wave Feshbach resonances have universal properties that are insensitive to the short-range details of the interaction. What is less known is that atoms near higher partial-wave Feshbach resonances also have remarkable universal properties. We illustrate this with a single atom interacting resonantly with two fixed static centers. At a Feshbach resonance point with orbital angular momentum $L\ensuremath{\ge}1$, we find 2L+1 shallow bound states whose energies behave like 1/R2L+1 when the distance R between the two centers is large. We then compute corrections to the binding energies due to other parameters in the effective range expansions. For completeness we also compute the binding energies near s-wave Feshbach resonances, taking into account the corrections. Afterwards we turn to the bound states at large but finite scattering volumes. For p-wave and higher partial-wave resonances we derive a simple formula for the energies in terms of a parameter called ``proximity parameter.'' These results are applicable to a free atom interacting resonantly with two atoms that are localized to two lattice sites of an optical lattice, and to one light atom interacting with two heavy ones in free space. Modifications of the low energy physics due to the long range van der Waals potential are also discussed.