^{1}and T. Stoecklin

^{1,a)}

### Abstract

We take advantage of the simple expression of the sector adiabatic wave functions of the Magnus propagator to obtain accurate values of the energy derivative of the matrix which, in turn, is used to get the Smith lifetime matrix. The procedure involves the simultaneous generation of both the matrix and its energy derivative which are propagated along the scattering coordinate. We present a few examples of application to the field free inelastic collisions which we previously studied. This method is then applied to the calculation of the lifetime of tuned zero energy Feshbach resonances using a magnetic field. We give and discuss the law of variation as a function of the magnetic field of the matrix eigenvalues across such resonance. Some examples of application are given for the collisions in a magnetic field.

I. INTRODUCTION

II. THEORY AND CALCULATIONS

III. APPLICATIONS TO THE collisions

A. Calculations for resonances occurring in field free ultracold inelastic collisions

B. Applications to zero energy Feshbach resonances tuning using a magnetic field

IV. CONCLUSION

### Key Topics

- Magnetic fields
- 18.0
- S matrix theory
- 17.0
- Magnetic resonance
- 16.0
- Eigenvalues
- 13.0
- Wave functions
- 5.0

## Figures

Comparison of the partial wave cross section and the largest eigenlifetime as a function of collision energy in , for the transition for the collision of with .

Comparison of the partial wave cross section and the largest eigenlifetime as a function of collision energy in , for the transition for the collision of with .

Comparison of the partial wave cross section and the largest eigenlifetime for as a function of collision energy in , for the transition for the collision of with .

Comparison of the partial wave cross section and the largest eigenlifetime for as a function of collision energy in , for the transition for the collision of with .

Comparison of the partial wave cross section and the largest eigenlifetime for as a function of collision energy in , for the transition during the collision of with .

Comparison of the partial wave cross section and the largest eigenlifetime for as a function of collision energy in , for the transition during the collision of with .

Comparison of the partial wave cross section and the largest eigenlifetime for as a function of collision energy in , for the transition during the collision of with .

Two upper panels: comparison of the partial wave cross section and the largest matrix eigenvalue for as a function of collision energy in , for the transition during the collision of with . The lower panel shows the three largest eigenvalues of the lifetime matrix (denoted el 1, el 2, and el 3) for the same partial wave.

Two upper panels: comparison of the partial wave cross section and the largest matrix eigenvalue for as a function of collision energy in , for the transition during the collision of with . The lower panel shows the three largest eigenvalues of the lifetime matrix (denoted el 1, el 2, and el 3) for the same partial wave.

(a) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of . (b) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of .

(a) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of . (b) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of .

(a) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of . (b) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of . (c) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of .

(a) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of . (b) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of . (c) Highest positive (denoted el 14) and lowest negative (denoted el 1) matrix eigenvalues for the elastic collision as a function of the magnetic field from calculations at a kinetic energy of .

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