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Theory of Complex Scattering Lengths

We derive a generalized Low equation for the T-matrix appropriate for complex atom-molecule interaction. The properties of this new equation at very low energies are studied and the complex scattering length and effective range are derived.

The recent realization of Bose-Einstein condensation (BEC) of ultracold atoms with the accompanying upsurge of theoretical activities have rekindled interest in low energy collisions of atoms and molecules. The subsequent proposals for the creation of ultracold molecular [1][2][3][4][5] and hybrid atomic-molecular BEC [6,7] intensified the above mentioned interest Of particular importance in the above recent developments is the idea of decay of the condensates. In a series of papers, Dalgarno and collaborators [8][9][10][11][12] have looked into the idea of using a complex scattering length to represent the low-energy atom-molecule scattering. Implicit in this is the multichannel nature of the collision process: an atom hits a vibrationally excited molecule at extremely low energies. The open inelastic channels are those where the molecule is excited into lower vibrational states. In this sense one has a depletion of the elastic channel. In Ref. [8], the quenching of H 2 molecules in collisions with H was considered. It was found that, the inelastic cross-sections and the corresponding depletion rate coefficients were very large for high vibrational levels of H 2 .
In the above studies, the following form of low-energy S-wave scattering amplitude is where k is the wave number related to the center of mass energy of the colliding partners, E, by ℏk 2 2µ = E, with µ being the reduced mass of the system. The function g (k 2 ) is even in k and is given by the effective range formula.
where a is the scattering length and r • the effective range, both directly related to the interaction. When applied to atom-molecule scattering at very low energies, with the molecules suffering inelastic transitions to lower vibrational states, the scattering length a is taken to be complex, a = α − iβ, with β related to the total inelastic cross-section.
The question we raise here is the validity of Eq. (1) with a and eventually r • taken as complex in the case of the elastic scattering with strong coupling to inelastic channels.
Of course, an equivalent one-channel description of the elastic scattering can be formulated with the introduction of an appropriate complex optical potential as described by Feshbach [13]. It is therefore legitimate to inquire about the validity of Eq. (1), originally obtained for real potential, if a complex interaction is used [14]. The general structure of the low energy scattering amplitude is also of potentially fundamental importance to very low energy matter interferometry. This method for the obtention of f for molecule-molecule scattering has been quite successful at room temperatures [15,16]. Extension to very low temperatures of this method seems natural and would welcome studies of the type reported here.
For the above purpose, it is useful to summarize the elegant derivation of Eq. (1) given by Weinberg [17]. If we denote the interaction by V and the free Green's function by

then the T-matrix given by the Lippmann-Schwinger equation
Using the spectral expansion of G (+) , with the complete set of bound and scattering states |B , Ψ (+) , we obtain the Low-equation At very low energies relevant for BEC, we seek a solution T k′k (E) ≡ T (E) and writing Accordingly, Eq. (3) now reads It is clear that the Low equation, Eq. (3), is not valid anymore. However, as we show below Eq. (1) is still valid, with the appropriate generalization of the real function g (k 2 ) to a complex one [16]. To see this we analyze the matrix element Ψ (+) k′′ |V | k . From the L − S equation for Ψ (+) k′′ , Thus Ψ (+) k′′ |V | k = k′ T (E k′′ − iε) k , where the unphysical T-matrixT is given by Accordingly the T-matrix equation, Eq (7), may be written as A similar equation holds for k′ T (E) k with iε replaced by −iε. It is interesting at this point to show the relation between the physical T-matrix element k′ |T (E)| k and k′ T (E) k . This can be done easily following operator manipulations of [18], and using the relation Ψ (+) where S −1 is the inverse S-matrix in the elastic channel, S −1 , and the diagonal part of the matrix element Ψ k′′ is directly related to the total inelastic scattering cross-section, σ in , viz [18] Eq. (11) explicitly exhibits the connection betweenT and T through the absorptive part of the effective interaction.
Now we seek the low energy solution k′ |T | k ≡ t + (E) and k′ T k ≡ t − (E) and following the same steps as Weinberg's [17], we find immediately, from Eq. (10), with where g c (k 2 ) is the complex generalization of g (k 2 ) of Eq. (1).
We turn now to the connection between g c (k 2 ) and the low-energy observables. This is most conveniently accomplished by employing the generalized optical theorem where σ el is the total elastic scattering cross section 4π |f + | 2 and σ in the total inelastic cross-section.
Using (12), we find At k = 0, g c (0) = − 1 a , where a is the complex scattering length written as [8] α − iβ. Thus the imaginary part of a, β, is found to be an expression also derived in Ref. [8]. Eq. (16) clearly implies that σ in should go as k −1 as k is lowered, in accordance with Wigner's law.
We go a bit beyond Refs. [8][9][10][11][12] and derive a relation between β and the imaginary part of the effective potential. Since σ in is given by (for S-wave scattering), Eq. (12) where u (r) is the S-wave elastic radial wave function, we find An equation for the complex effective range, r • , can also be easily derived. Equation (19) is the principle result of this work. It summarizes the following: 1) The coupled-channels calculation aimed to describe the molecular quenching can be recast as an effective one-channel calculation with a complex interaction whose imaginary part account for flux loss.
2) The low-energy behaviour of the scattering amplitude with the complex interaction alluded to above can be conveniently parametrized in terms of complex scattering length and effective range.
The message this work conveys is the potential usefulness of constructing the effective complex (optical) interaction for the scattering of ro-vibrational molecules from atoms at low energies. The calculation of a and r • from knowledge of this potential can be done in a direct and unambiguous way.