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arXiv:1107.1006 · Journal of Physics B Atomic Molecular and Optical Physics

Benchmark helium dimer and trimer calculations with a public few-body code

Vladimir Roudnev, M. J. Cavagnero

Published Dec 16, 2011• 27 citations
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Abstract

We present detailed calculations of bound and scattering states of dimers and trimers of He to produce highly accurate data and to test a non-relativistic three-body code currently in development for public distribution. For these systems, uncertainties and inaccuracies in the fundamental constants frequently used in published works can substantially exceed numerical errors. Our benchmark calculations include specific estimates of the numerical accuracy of the calculations and also explore sensitivity to fundamental constants and their uncertainties. The use of an inexact coupling constant in the previous calculations leads to 0.08% error for the ground state energy, 0.3% error for the excited state energy and up to 0.15% error for the atom–dimer scattering length in the system of three 4 He atoms. The corresponding errors for the asymmetric 4 He 2 3 He system are 0.3% for the bound state energy and 0.03% for the atom–dimer scattering length.

DOI
10.1088/0953-4075/45/2/025101

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Cold helium atoms are a unique system in which a single excited three-body Efimov bound state occurs, naturally, without the need for an external magnetic field. While three-body bound-state energies of cold helium atoms have previously been investigated, recent experimental techniques have allowed their structure to also be studied. The weak interaction between helium atoms leads to a helium-helium (dimer) scattering length $a$ much larger than the helium-helium effective range of interaction $r$. This feature is exploited in a theory that systematically expands observables in powers of $r/a$, known as short-range effective field theory (SREFT), which has been used successfully to investigate properties of cold-atom systems. Using SREFT, we investigate the average bond length of atoms in the three-body ground state and excited Efimov bound state of cold helium atoms. At leading-order (next-to-leading order) in SREFT, we find the average bond length of the $^{4}\mathrm{He}$ trimer ground state is 8.35(33) $\AA{}$ (10.29(2) $\AA{}$) and the average bond length of the excited $^{4}\mathrm{He}$ trimer Efimov bound state is 103(4) $\AA{}$ (105.3(2) $\AA{}$).

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We present detailed calculations of bound and scattering states of dimers and trimers of He to produce highly accurate data and to test a non-relativistic three-body code currently in development for public distribution. For these systems, uncertainties and inaccuracies in the fundamental constants frequently used in published works can substantially exceed numerical errors. Our benchmark calculations include specific estimates of the numerical accuracy of the calculations and also explore sensitivity to fundamental constants and their uncertainties. The use of an inexact coupling constant in the previous calculations leads to 0.08% error for the ground state energy, 0.3% error for the excited state energy and up to 0.15% error for the atom–dimer scattering length in the system of three 4 He atoms. The corresponding errors for the asymmetric 4 He 2 3 He system are 0.3% for the bound state energy and 0.03% for the atom–dimer scattering length.
Four-body system of 4He{}^4\mathrm{He}4He atoms: Dimer-dimer scattering
The strong short-range repulsion, characteristic to realistic interatomic potentials, complicates the description of weakly-bound few-body systems such as those of \He atoms. The present work proposes an approach for solving this problem and applies it to a realistic system of four 4He{}^4\mathrm{He}4He atoms. The potential is gradually softened such that rigorous four-body equations for bound and scattering tates can be accurately solved in the momentum-space framework, and the results are extrapolated back to the limit of the original potential. Linear correlations between three- and four body quantities are observed, and the accuracy of the procedure is improved by extrapolating in one of the three-body quantities. Results for the 4He{}^4\mathrm{He}4He tetramer ground and excited state binding energies and atom-trimer scattering agree well with at least some of earlier determinations and shed light on the existing disagreements. An additional case of the Phillips correlation line is established for the dimer-dimer scattering length. The trimer production rate via the ultracold two-dimer collisions is estimated, it exhibits significant finite-range effects despite the weak binding of the dimer.
We present an illustration of using a quantum three-body code being prepared for public release. The code is based on iterative solving of the three-dimensional Faddeev equations. The code is easy to use and allows users to perform highly-accurate calculations of quantum three-body systems. The previously known results for He3_33​ ground state are well reproduced by the code.
The two-body scattering length a for 4He{}^{4}\mathrm{He}4He atoms is much larger than their effective range rs.{r}_{s}.rs​. As a consequence, low-energy few-body observables have universal characteristics that are independent of the interaction potential. Universality implies that, up to corrections suppressed by rs/a,{r}_{s}/a,rs​/a, all low-energy three-body observables are determined by a and a three-body parameter ${\ensuremath{\Lambda}}_{*}.$ We give simple expressions in terms of a and ${\ensuremath{\Lambda}}_{*}$ for the trimer binding-energy equation, the atom-dimer scattering phase shifts, and the rate for three-body recombination at threshold. We determine ${\ensuremath{\Lambda}}_{*}$ for several 4He{}^{4}\mathrm{He}4He potentials from the calculated binding energy of the excited state of the trimer and use it to obtain the universality predictions for the other low-energy observables. We also use the calculated values for one potential to estimate the effective range corrections for the other potentials.
Small Clusters of 4^44He Atoms in Finite-Cutoff Effective Field Theory
Small clusters of 4^44He atoms are benchmark systems for universal few-body physics near the unitary limit. We study these systems using a finite-cutoff effective field theory calibrated to low-energy observables from the realistic LM2M2 potential. The chosen two-body cutoff reproduces both the atom--atom scattering length and effective range, while a regulated three-body interaction is adjusted to the trimer and tetramer ground-state energies. We distinguish total binding energies from the one-atom separation energies SA∗=BA∗−BA−1S_A^*=B_A^*-B_{A-1}SA∗​=BA∗​−BA−1​ of the shallow excited states. Ground-state energies through A=8A=8A=8 are reproduced at the few-percent level, but threshold-sensitive observables are less accurate. In particular, the calculated tetramer separation energy is S4∗=2.85S_4^*=2.85S4∗​=2.85~mK, compared with 0.920.920.92--0.960.960.96~mK in converged LM2M2 calculations, and the atom--trimer scattering length differs substantially from modern LM2M2 benchmarks. For A≥5A\ge5A≥5, the available excited-state benchmarks are sparse and strongly method dependent. Comparison with another soft-core interaction suggests that the discrepancies primarily reflect short-distance physics and higher-order operators omitted from the finite-cutoff Hamiltonian rather than numerical convergence of the stochastic variational calculation. Finite-cutoff EFT therefore provides an efficient description of ground-state systematics, whereas shallow excited states and atom--cluster scattering require a controlled higher-order and cutoff-dependence analysis.
In a previous work [Phys. Rev. A 85, 022502 (2012)] we calculated, with the use of the Gaussian expansion method for few-body systems, the energy levels and spatial structures of the 4{}^{4}4He trimer and tetramer ground and excited states using the LM2M2 potential, which has a very strong short-range repulsion. In this work, we calculate the same quantities using the currently most accurate 4{}^{4}4He-4{}^{4}4He potential [Przybytek et al., Phys. Rev. Lett. 104, 183003 (2010)] that includes the adiabatic, relativistic, QED, and residual retardation corrections. Contributions of the corrections to the tetramer ground-(excited-) state energy $\ensuremath{-}573.90\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}132.70)$ mK are respectively $\ensuremath{-}4.13\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}1.52)$ mK, +9.37(+3.48)+9.37\phantom{\rule{0.16em}{0ex}}(+3.48)+9.37(+3.48) mK, $\ensuremath{-}1.20\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}0.46)$ mK, and +0.16(+0.07)+0.16\phantom{\rule{0.16em}{0ex}}(+0.07)+0.16(+0.07) mK. Further including other realistic 4{}^{4}4He potentials, we calculated the binding energies of the trimer and tetramer ground and excited states, B3(0),B3(1),B4(0){B}_{3}^{(0)},\phantom{\rule{0.16em}{0ex}}{B}_{3}^{(1)},\phantom{\rule{0.16em}{0ex}}{B}_{4}^{(0)}B3(0)​,B3(1)​,B4(0)​, and B4(1){B}_{4}^{(1)}B4(1)​, respectively. We found that the four kinds of the binding energies for the different potentials exhibit perfect linear correlations between any two of them over the range of binding energies relevant for 4{}^{4}4He atoms (namely, six types of the generalized Tjon lines are observed). The dimerlike-pair model for 4{}^{4}4He clusters, proposed in the previous work, predicts a simple interaction-independent relation B4(1)B2=B3(0)B2+23\frac{{B}_{4}^{(1)}}{{B}_{2}}=\frac{{B}_{3}^{(0)}}{{B}_{2}}+\frac{2}{3}B2​B4(1)​​=B2​B3(0)​​+32​, which precisely explains the correlation between the tetramer excited-state energy and the trimer ground-state energy, with B2{B}_{2}B2​ being the dimer binding energy.
Cold helium atoms are a unique system in which a single excited three-body Efimov bound state occurs, naturally, without the need for an external magnetic field. While three-body bound-state energies of cold helium atoms have previously been investigated, recent experimental techniques have allowed their structure to also be studied. The weak interaction between helium atoms leads to a helium-helium (dimer) scattering length aaa much larger than the helium-helium effective range of interaction rrr. This feature is exploited in a theory that systematically expands observables in powers of r/ar/ar/a, known as short-range effective field theory (SREFT), which has been used successfully to investigate properties of cold-atom systems. Using SREFT, we investigate the average bond length of atoms in the three-body ground state and excited Efimov bound state of cold helium atoms. At leading-order (next-to-leading order) in SREFT, we find the average bond length of the 4He^{4}\mathrm{He}4He trimer ground state is 8.35(33) A˚\AA{}A˚ (10.29(2) A˚\AA{}A˚) and the average bond length of the excited 4He^{4}\mathrm{He}4He trimer Efimov bound state is 103(4) A˚\AA{}A˚ (105.3(2) A˚\AA{}A˚).