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arXiv:1708.04661 · Physical Review A

Characterizing Feshbach resonances in ultracold scattering calculations

Matthew D. Frye, Jeremy M. Hutson

Published Oct 11, 2017• 19 citations
arXivHTMLPDF

Abstract

We describe procedures for converging on and characterizing zero-energy Feshbach resonances that appear in scattering lengths for ultracold atomic and molecular collisions as a function of an external field. The elastic procedure is appropriate for purely elastic scattering, where the scattering length is real and displays a true pole. The regularized scattering length procedure is appropriate when there is weak background inelasticity, so that the scattering length is complex and displays an oscillation rather than a pole, but the resonant scattering length ${a}_{\mathrm{res}}$ is close to real. The fully complex procedure is appropriate when there is substantial background inelasticity and the real and imaginary parts of ${a}_{\mathrm{res}}$ are required. We demonstrate these procedures for scattering of ultracold $^{85}\mathrm{Rb}$ in various initial states. All of them can converge on and provide full characterization of resonances, from initial guesses many thousands of widths away, using scattering calculations at only about ten values of the external field.

DOI
10.1103/physreva.96.042705

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We describe procedures for converging on and characterizing zero-energy Feshbach resonances that appear in scattering lengths for ultracold atomic and molecular collisions as a function of an external field. The elastic procedure is appropriate for purely elastic scattering, where the scattering length is real and displays a true pole. The regularized scattering length procedure is appropriate when there is weak background inelasticity, so that the scattering length is complex and displays an oscillation rather than a pole, but the resonant scattering length ares{a}_{\mathrm{res}}ares​ is close to real. The fully complex procedure is appropriate when there is substantial background inelasticity and the real and imaginary parts of ares{a}_{\mathrm{res}}ares​ are required. We demonstrate these procedures for scattering of ultracold 85Rb^{85}\mathrm{Rb}85Rb in various initial states. All of them can converge on and provide full characterization of resonances, from initial guesses many thousands of widths away, using scattering calculations at only about ten values of the external field.
In the absence of inelastic scattering, Feshbach resonances produce poles in scattering lengths and very large peaks in elastic cross-sections. However, inelastic scattering removes the poles. Whenever the resonant state is coupled comparably to the elastic and inelastic channels, the scattering length exhibits only a small oscillation and peaks in cross-sections are significantly suppressed. A resonant scattering length is defined to characterize the amplitude of the oscillation, and is shown to be small for many collisions of ultracold molecules. The results suggest that cross-sections for some ultracold collision processes will be much less sensitive to details of the potential than has been expected.
We have generalized the BOUND and MOLSCAT packages to allow calculations in basis sets where the monomer Hamiltonians are off diagonal and used this capability to carry out bound-state and scattering calculations on $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{NH}$ and $^{4}\mathrm{He}\text{\ensuremath{-}}\mathrm{NH}$ as a function of magnetic field. Following the bound-state energies to the point where they cross thresholds gives very precise predictions of the magnetic fields at which zero-energy Feshbach resonances occur. We have used this to locate and characterize two very narrow Feshbach resonances in $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{NH}$. Such resonances can be used to tune elastic and inelastic collision cross sections, and sweeping the magnetic field across them will allow a form of quantum control in which separated atoms and molecules are associated to form complexes. For the first resonance, where only elastic scattering is possible, the scattering length shows a pole as a function of magnetic field and there is a very large peak in the elastic cross section. For the second resonance, however, inelastic scattering is also possible. In this case the pole in the scattering length is dramatically suppressed and the cross sections show relatively small peaks. The peak suppression is expected to be even larger in systems with stronger inelasticity. The results suggest that calculations on ultracold molecular inelastic collisions may be much less sensitive to details of the potential energy surface than has been believed.
This paper investigates the time delay around Feshbach resonances in ultracold collisions. Far above threshold, the time delay shows a simple Lorentzian peak as a function of both energy and field, but close to threshold it is proportional to the scattering length and so has a pole-like oscillation as a function of field. For narrow resonances, the energy of the crossover between these behaviors is proportional to the square of the resonance strength sres{s}_{\text{res}}sres​. For resonances that are wide or have large background scattering lengths, the behavior is more complicated.
We have carried out bound-state and low-energy quantum scattering calculations on He + NH (triplet Sigma) in magnetic fields, with the NH molecule in its n=1 rotationally excited states. We have explored the pattern of levels as a function of magnetic field and identified the nearly good quantum numbers in different regimes. We have used the bound-state calculations to locate low-energy Feshbach resonances. When the magnetic field is used to tune across such a resonance, the real and imaginary part of the scattering length show asymmetric oscillations and peaks with amplitude between 1 and 3 Angstroms. The scattering length does not pass through a pole at resonance. The resonant behavior is characterized by a complex resonant scattering length a_res. The corresponding inelastic cross sections show troughs as well as peaks near resonance. This may be important for efforts to achieve evaporative and sympathetic cooling for molecules, because it offers the hope that inelastic trap losses can be reduced by tuning close to a Feshbach resonance.
The s-wave scattering length for elastic collisions between 87Rb{}^{87}\mathrm{Rb}87Rb atoms in the state ∣f,mf〉=∣1,1〉|{f,m}_{f}〉=|1,1〉∣f,mf​〉=∣1,1〉 is measured in the vicinity of a Feshbach resonance near 1007 G. Experimentally, the scattering length is determined from the mean-field-driven expansion of a Bose-Einstein condensate in a homogeneous magnetic field. The scattering length is measured as a function of the magnetic field and agrees with the theoretical expectation. The width of the resonance is determined to be 0.20(3) G, the position of the zero crossing of the scattering length is found at 1007.60(3) G.