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Physical Review A

Low-Energy Atom-Atom Scattering: Corrections to the He-He Interaction

Marvin H. Mittleman, Hsiang Tai

Published Oct 1, 1973• 35 citations
Publisher

Abstract

A formalism, previously presented, for describing atom-atom scattering in the adiabatic representation to higher orders in $\frac{m}{M}$ is generalized to describe the problem of different isotopes and different atoms. This is accomplished by a sequence of nonlinear coordinate transformations which introduce a "switching function" which is defined only in the asymptotic limit of separated atoms. It is found that the effective potential for elastic scattering contains energy-dependent terms in order $\frac{m}{M}$ and that it also depends upon the switching function in this order so that the potential is not unique. It is also found that in this order the definition of adiabatic states is not a trivial problem. A variational choice is made for the switching function and explicit calculations are presented for the He-He interaction. The coupled equations describing transitions are then derived and it is shown that terms are omitted from the usual coupling operator which are of the same order of magnitude as the one usually kept. This leads to a modification of the usual Landau-Zener transition formula.

DOI
10.1103/physreva.8.1880

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arXiv:0909.0834•Laser Physics•Dec 7, 2009•10 citations
A formalism, previously presented, for describing atom-atom scattering in the adiabatic representation to higher orders in mM\frac{m}{M}Mm​ is generalized to describe the problem of different isotopes and different atoms. This is accomplished by a sequence of nonlinear coordinate transformations which introduce a "switching function" which is defined only in the asymptotic limit of separated atoms. It is found that the effective potential for elastic scattering contains energy-dependent terms in order mM\frac{m}{M}Mm​ and that it also depends upon the switching function in this order so that the potential is not unique. It is also found that in this order the definition of adiabatic states is not a trivial problem. A variational choice is made for the switching function and explicit calculations are presented for the He-He interaction. The coupled equations describing transitions are then derived and it is shown that terms are omitted from the usual coupling operator which are of the same order of magnitude as the one usually kept. This leads to a modification of the usual Landau-Zener transition formula.
The present theory of low-energy scattering by a long-range potential has been examined by omitting the assumptions about the wavefunctions and taking the phase to be a function of the position of the scattered particle. The resulting integro-differential equation for the phase shift has been simplified for a particular case and it is found that the magnitude of the correction terms so obtained is quite large. It is shown that the two-potential formalism can be applied only to light pairs of scattering atoms.
An energy-scaling rule for atom-atom scattering is derived. It is shown to be approximately accurate to third order in the parameter $\frac{\ensuremath{\Delta}M}{{M}_{\mathrm{av}}}$, where $\ensuremath{\Delta}M$ is the difference in the reduced masses between two different atomic pairs. The energy scaling depends upon the energy itself and on the interaction potential. The method is illustrated by the He-He scattering problem using the isotopes He3{\mathrm{He}}_{3}He3​ and He4{\mathrm{He}}_{4}He4​ in their three different combinations.
An important feature of cold atomic vapours is that particle separations, which are typically of order 10 2 nm, are usually an order of magnitude larger than the length scales associated with the atom–atom interaction. Consequently, the two-body interaction between atoms dominates, and three-and higher-body interactions are unimportant. Moreover, since the atoms have low velocities, many properties of these systems may be calculated in terms of a single parameter, the scattering length. An alkali atom in its electronic ground state has several different hyperfine states, as we have seen in Secs. 3.1 and 3.2. Interatomic interactions give rise to transitions between these states and, as we described in Sec. 4.6, such processes are a major mechanism for loss of trapped atoms. In a scattering process, the internal states of the particles in the initial or final states are described by a set of quantum numbers, such as those for the spin, the atomic species, and their state of excitation. We shall refer to a possible choice of these quantum numbers as a channel . At the temperatures of interest for Bose–Einstein condensation, atoms are in their electronic ground states, and the only relevant internal states are therefore the hyperfine states. Because of the existence of several hyperfine states for a single atom, the scattering of cold alkali atoms is a multi-channel problem.
A variational procedure is developed for the calculation of the multiphoton-ionization rate of an atom interacting with a superposition of two time-dependent electric fields, one weak and the other intense and slowly varying. Trial functions are chosen that take into account the effect of both the atomic binding potential and the external fields. The variational expression contains, as its leading term, an approximation for the transition amplitude that was derived previously. The correction term is carefully analyzed here with regard to its dependence on suitably defined frequency and intensity parameters. The analysis provides firmly based estimates of the order, relative to the leading term, of the error in the approximation. A sum rule is derived which, owing to the cancellation of higher-order correction terms, allows for an accurate representation, in a relatively simple form, of the total transition rate summed over the net number of photons exchanged with the low-frequency field. A modified final-state trial function is presented which may be used, without loss of accuracy, to account for the existence of scattering resonances with energies close to that of the emergent electron. The calculational procedure is tested numerically on a one-dimensional model problem; the results are consistent with the more formal analysis of the relative order of the correction term and the accuracy of the sum rule.
In the framework of a basic semiclassical time-dependent nonlinear two-state problem, we study the weak coupling limit of the nonlinear Landau-Zener transition at coherent photo- and magneto-association of an atomic Bose-Einstein condensate. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct an accurate analytic approximation describing time dynamics of the coupled atom-molecular system for the case of weak coupling. The approximation is written in terms of the solution to an auxiliary linear Landau-Zener problem with some effective Landau-Zener parameter. The dependence of this effective parameter on the input Landau-Zener parameter is found to be unexpected: as the generic Landau-Zener parameter increases, the effective Landau-Zener parameter first monotonically increases (starting from zero), reaches its maximal value and then monotonically decreases again reaching zero at some point. The constructed approximation quantitatively well describes many characteristics of the time dynamics of the system, in particular, it provides a highly accurate formula for the final transition probability to the molecular state. The present result for the final transition probability improves the accuracy of the previous approximation by Ishkhanyan et al. [Phys. Rev. A 69 , 043612 (2004); J. Phys. A 38 , 3505 (2005)] by order of magnitude.