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arXiv:physics/0612242 · Physical Review A

Calculation of three-body resonances using slow-variable discretization coupled with a complex absorbing potential

Juan Blandón, Viatcheslav Kokoouline, Françoise Masnou-Seeuws

Published Apr 19, 2007• 29 citations
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Abstract

We developed a method to calculate positions and widths of three-body resonances. The method combines the hyperspherical adiabatic approach, slow variable discretization method [O. I. Tolstikhin et al., J. Phys. B 29, L389 (1996)], and a complex absorbing potential. The method can be used to obtain resonances having short-range or long-range wave functions. In particular, we have applied the method to obtain very shallow three-body Efimov resonances for a model system [E. Nielsen et al., Phys. Rev. A 66, 012705 (2002)].

DOI
10.1103/physreva.75.042508

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We developed a method to calculate positions and widths of three-body resonances. The method combines the hyperspherical adiabatic approach, slow variable discretization method [O. I. Tolstikhin et al., J. Phys. B 29, L389 (1996)], and a complex absorbing potential. The method can be used to obtain resonances having short-range or long-range wave functions. In particular, we have applied the method to obtain very shallow three-body Efimov resonances for a model system [E. Nielsen et al., Phys. Rev. A 66, 012705 (2002)].
In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the three-body system, (2) the solution of the Schr\"odinger equation in two steps: determination of adiabatic states for a fixed hyper-radius and then solution of a set of coupled hyper-radial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer 4He3^{4}\mathrm{He}_{3}4He3​, isotopomers of the H3+\mathrm{H}_{3}{}^{+}H3​+ ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.
The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.
We have performed a three-body calculation for investigation of S-wave resonances in the positron–helium scattering in the hyperspherical coordinates. B-splines are used to obtain accurate hyperspherical potential curves and channel functions. An adiabatical hyperspherical approach is used to predict semi-quantitatively the position of the resonances. A multichannel calculation with the stabilization method is also used to extract accurately the positions and widths of resonances. We find four resonances associated with the He + –Ps( n = 2) channel. The resonance parameters for the lowest two resonances are in reasonable agreement with the calculations of Kar and Ho ( 2004 J. Phys. B: At. Mol. Opt. Phys. 37 3177 ), and two more new resonances below the Ps( n = 2) threshold are identified.
We are developing theoretical and computational methods to study two related three-body processes in ultracold quantum gases: three-body resonances and three-body recombination. Three-body recombination causes the ultracold gas to heat up and atoms to leave the trap where they are confined. Therefore, it is an undesirable effect in the process of forming ultracold quantum gases. Metastable three-body states (resonances) are formed in the ultracold gas. When decaying they also give additional kinetic energy to the gas, that leads to the heating too. In addition, a reliable method to obtain three-body resonances would be useful in a number of problems in other fields of physics, for example, in models of metastable nuclei or to study dissociative recombination of H3 +. Our project consists of employing computer modeling to develop a method to obtain three-body resonances. The method uses a novel two-step diagonalization approach to solve the three-body Schrödinger equation. The approach employs the SVD method of Tolstikhin et al. coupled with a complex absorbing potential. We tested this method on a model system of three identical bosons with nucleon mass and compared it to the results of a previous study. This model can be employed to understand the 3He nucleus . We found one three-body bound state and four resonances. We are also studying Efimov resonances using a 4He-based model. In a system of identical spinless bosons, Efimov states are a series of loosely bound three-body states which begin to appear as the energy of the two-body bound state approaches zero . Although they were predicted 35 years ago, recent evidence of Efimov states found by Kraemer et al. in a gas of ultracold Cs atoms has sparked great interest by theorists and experimentalists. Efimov resonances are a kind of pre-dissociated Efimov trimer. To search for Efimov resonances we tune the diatom interaction potential, V(r): V(r) → λV(r) as Esry et al. did . We calculated the first two values of λ for which there is a "condensation" (infinite number) of Efimov states. They are λEfimov1 = 0.9765 and λEfimov2 = 6.834. We performed calculations for λ = 2.4, but found no evidence of Efimov resonances. For future work we plan to work with λ ≈ 4 and λ ≈ λEfimov2 where we might see d-wave and higher l-wave Efimov resonances. There is also a many-body project that forms part of this thesis and consists of a direct diagonalization of the Bogolyubov Hamiltonian, which describes elementary excitations of a gas of bosons interacting through a pairwise interaction. We would like to reproduce the corresponding energy spectrum. So far we have performed several convergence tests, but have not observed the desired energy spectrum. We show preliminary results.
Quantum mechanical calculations of ultralow-energy collision of Coulomb three-body systems in the hyperspherical elliptic coordinates are presented. The nonadiabatic coupling between the hyperradius and hyperangular variables is treated with the slow-variable discretization method in combination with the RRR-matrix propagation technique. For scattering state calculations, the two-dimensional matching procedure using Gailitis's method [M. Gailitis, J. Phys. B 9, 843 (1976); C. Noble and R. Nesbet, Comput. Phys. Commun. 33, 399 (1984)] is implemented to determine the boundary conditions between the internal and the asymptotic wave functions. This method is proved to be very efficient and gives very accurate results. Taking advantage of this method, we accurately calculate the scattering phase shifts and the scattering lengths of Coulomb three-body systems with mass ratio varying over several orders of magnitudes. We observed jumps of the scattering length from $\ensuremath{-}\ensuremath{\infty}$ to $\ensuremath{\infty}$ at certain mass ratios and monotonic decreases between two jumps. These are closely related to the binding energy of the highest bound state through Levinson's theorem [N. Levinson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 25, 9 (1949)]. Our calculations provide a comprehensive perspective to the scattering length from the variation of mass ratio of Coulomb three-body systems.
Hyperspherical coordinates provide a systematic way of describing three-body systems. Solving three-body Schrödinger equations in an adiabatic hyperspherical representation is the focus of this thesis. An essentially exact solution can be found numerically by including nonadiabatic couplings using either a slow variable discretization or a traditional adiabatic method. Two different types of three-body systems are investigated: (1) rovibrational states of the triatomic hydrogen ion H+3 and (2) ultracold collisions of three identical bosons.