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arXiv:0709.0343 · Physical Review A

Exactly solvable coupled-channel potential models of atom-atom magnetic Feshbach resonances from supersymmetric quantum mechanics

Andrey M. Pupasov-Maksimov, B. F. Samsonov, Jean-Marc Sparenberg

Published Jan 30, 2008• 14 citations
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Abstract

Starting from a system of $N$ radial Schr\"odinger equations with a vanishing potential and finite threshold differences between the channels, a coupled $N\ifmmode\times\else\texttimes\fi{}N$ exactly solvable potential model is obtained with the help of a single nonconservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as does its Jost matrix. It depends on $N(N+1)/2$ unconstrained parameters and on one upper-bounded parameter, the factorization energy. A detailed study of the model is done for the $2\ifmmode\times\else\texttimes\fi{}2$ case: a geometrical analysis of the zeros of the Jost-matrix determinant shows that the model has zero, one, or two bound states, and zero or one resonance; a compact formula for the open-channel scattering length is obtained. As a first physical application, exactly solvable $2\ifmmode\times\else\texttimes\fi{}2$ atom-atom interaction potentials are constructed, for cases where a magnetic Feshbach resonance interacts with a bound or virtual state close to threshold, which results in a large background scattering length.

DOI
10.1103/physreva.77.012724

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Starting from a system of NNN radial Schr"odinger equations with a vanishing potential and finite threshold differences between the channels, a coupled N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN exactly solvable potential model is obtained with the help of a single nonconservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as does its Jost matrix. It depends on N(N+1)/2N(N+1)/2N(N+1)/2 unconstrained parameters and on one upper-bounded parameter, the factorization energy. A detailed study of the model is done for the 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 case: a geometrical analysis of the zeros of the Jost-matrix determinant shows that the model has zero, one, or two bound states, and zero or one resonance; a compact formula for the open-channel scattering length is obtained. As a first physical application, exactly solvable 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 atom-atom interaction potentials are constructed, for cases where a magnetic Feshbach resonance interacts with a bound or virtual state close to threshold, which results in a large background scattering length.
We discuss sss-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.
Relations between propagators and Green functions of Hamiltonians which are SUSY partners have been obtained. New exact propagators for the family of multi-well, time-dependent and non-hermitian potentials have been calculated. Non-conservative SUSY transformation has been studied in the case of multichannel Schrodinger equation with different thresholds. Spectrum (bound/virtual states and resonances) of the non-conservative SUSY partner of zero potential has been founded. Exactly solvable model of the magnetic induced Feshbach resonance has been constructed. This model was tested in the case of Rb85^{85}85. Conservative SUSY transformations of the first and the second order have been studied in the case of multichannel Schrodinger equation with equal thresholds. Transformations which introduce non-trivial coupling between scattering channels have been founded. The first order SUSY transformation which preserves SSS-matrix eigen-phase shifts and modifies mixing parameter has been founded in the case of two channel scattering with partial waves of different parities. In the case of coinciding parities we have found the second order SUSY transformation which preserves SSS-matrix eigen-phase shifts and modifies mixing parameter. Phenomenological two channel 3S1−3D1^3S_1-^3D_13S1​−3D1​ neutron-proton potential has been obtained by using single channel, phase equivalent and coupling SUSY transformations applied to zero potential.
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).
A Feshbach resonance arises in cold atom scattering due to the complex interplay between several coupled channels. However, the essential physics of the resonance may be encapsulated in a simplified model consisting of just two coupled channels. In this paper we describe in detail how such an effective Feshbach model can be constructed from knowledge of a few key parameters, characterizing the atomic Born-Oppenheimer potentials and the low energy scattering near the resonance. These parameters may be obtained either from experiment or full coupled-channel calculations. Using RRR-matrix theory we analyze the bound state spectrum and the scattering properties of the two-channel model, and find it to be in good agreement with exact calculations.
Abstract 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 them, the potential resonances at vanishing energy, occurs as a bound state of the coupled system emerges, in turn, 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.
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^{85}\mathrm{Rb}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^{6}\mathrm{Li}6Li and 133Cs^{133}\mathrm{Cs}133Cs. Our approach provides full insight into the underlying physics of the interplay between open-channel (or potential) resonances and Feshbach resonances.