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The Journal of Chemical Physics

Analytical calculation of the Smith lifetime Q matrix using a Magnus propagator: Applications to the study of resonances occurring in ultracold inelastic collisions with and without an applied magnetic field

Grégoire Guillon, Thierry Stoecklin

Published Apr 9, 2009• 16 citations
Publisher

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 S matrix which, in turn, is used to get the Smith lifetime Q matrix. The procedure involves the simultaneous generation of both the R matrix and its energy derivative dR/dE which are propagated along the scattering coordinate. We present a few examples of application to the field free He-N(2)(+) 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 Q matrix eigenvalues across such resonance. Some examples of application are given for the He-N(2)(+) collisions in a magnetic field.

DOI
10.1063/1.3111881

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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 S matrix which, in turn, is used to get the Smith lifetime Q matrix. The procedure involves the simultaneous generation of both the R matrix and its energy derivative dR/dE which are propagated along the scattering coordinate. We present a few examples of application to the field free He-N(2)(+) 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 Q matrix eigenvalues across such resonance. Some examples of application are given for the He-N(2)(+) collisions in a magnetic field.
We give a brief account of the effect of a magnetic field on the spin-depolarization cross sections of N2 +(2Σ) in collisions with 3He at very low collision energy. The fundamental states of the two nuclear spin isomers of N2 + are shown to respond quite differently to the application of the field. When the applied magnetic field increases, the spin depolarization cross sections are found to monotonously decrease for the fundamental para level N=0 of N2 +, whereas they monotonously increase for the fundamental ortho level N=1 of N2 +. This effect is found to be amplified around the resonances and an explanation based on the different mechanism of spin flipping acting for these two levels is proposed. We also explore the tuning of zero energy Feshbach resonances by examining the variation of the scattering length as a function of the magnetic field. We use the Smith Q matrix to calculate the lifetimes of the complexes associated with these resonances and discuss the law of variation as a function of the magnetic field of the Q matrix eigenvalues across such resonances.
We study the variation of the positions of two magnetically tuned Zero energy Feshbach resonances when a parallel superimposed electric field is applied. We show that their variation as a function of the electric field follows a simple analytical law and is then predictable. We find that depending on the initial state of the diatomic molecule the resonance is either shifted to higher or to lower values of the magnetic field when the electric field is applied. We calculate the Close Coupling lifetimes of these resonances and show that they also follow a simple law as a function of both the magnetic and the electric field. We demonstrate using this expression that the lifetime of the resonances can be maximised by choosing an appropriate value of the applied electric and found a good agreement with the results of our Close Coupling calculations. These results could be checked in future experiments dedicated to the 3He + NH collisions
We study the variation in the positions of two magnetically tuned zero energy Feshbach resonances when a parallel superimposed electric field is applied. We show that their variation as a function of the electric field follows a simple analytical law and is then predictable. We find that depending on the initial state of the diatomic molecule the resonance is either shifted to higher or to lower values of the magnetic field when the electric field is applied. We calculate the close-coupling lifetimes of these resonances and show that they also follow a simple law as a function of both the magnetic and the electric field. We demonstrate using this expression that the lifetime of the resonances can be maximized by choosing an appropriate value of the applied electric field and found a good agreement with the results of our close-coupling calculations. These results could be checked in future experiments dedicated to the $^{3}\text{He}+\text{NH}(^{3}\ensuremath{\Sigma})$ collisions
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.
While collision lifetimes are a fundamental property of few-body scattering events, their behavior at ultralow temperatures is not completely understood. Using multichannel quantum-defect theory, we derive a multichannel expression for the Smith lifetime QQQ matrix that is valid far away from scattering resonances. For collisions mediated by short-range interaction potentials of the form $\ensuremath{-}{C}_{6}/{R}^{6}$, we obtain the average time delay in the incident sss-wave collision channel in the limit of zero collision energy EEE as Q11=A/E+B{Q}_{11}=A/\sqrt{E}+BQ11​=A/E​+B, where AAA and BBB are constants. We show that the time delay is dominated by elastic scattering and contains an additional multichannel contribution independent of the two-body scattering length. We also obtain the expressions for the collision lifetime using the universal model of Idziaszek and Julienne [Phys. Rev. Lett. 104, 113202 (2010)]. The lifetime acquires an imaginary part in the presence of inelastic loss due to the lack of unitarity of the SSS matrix and shortens with increasing the short-range loss parameter yyy.
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.