arXiv:1409.2501 · Physical Review A

Universality in Efimov-associated tetramers in He4

Emiko Hiyama, M. Kamimura

Published Nov 18, 2014• 30 citations

Abstract

We calculated, using seven realistic $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials in the literature, the Efimov spectra of the $^{4}\mathrm{He}$ trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schr\"odinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the $^{4}\mathrm{He}$ trimer and tetramer [Phys. Rev. A 85, 022502 (2012); Phys. Rev. A 85, 062505 (2012)]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length ${r}_{\mathrm{vdW}}$, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius ${R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius ${R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$ makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the $^{4}\mathrm{He}$ trimer with the realistic potentials.

DOI: 10.1103/physreva.90.052514

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Linear correlations between 4 He trimer and tetramer energies calculated with various realistic 4 He potentials

Emiko Hiyama, M. Kamimura

arXiv:1203.3130Physical Review AJun 6, 201252 citations

Universality in Systems of up to Four Particles — From Nuclei to Ultracold Atoms

Christiane H. Schmickler

Jan 1, 20200 citations

Zero-energy scattering and bound states in the He4 trimer and tetramer

S. Nakaichi-Maeda, T. K. Lim

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Prediction of a weakly bound excited state of Efimov character in a Li7He24 system

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Universality and the three-body parameter of 4 He trimers

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arXiv:1409.2501 · Physical Review A

Universality in Efimov-associated tetramers in He4

Emiko Hiyama, M. Kamimura

Published Nov 18, 2014• 30 citations

arXiv HTML PDF

Abstract

We calculated, using seven realistic

^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}

potentials in the literature, the Efimov spectra of the

^{4}\mathrm{He}

trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schr"odinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the

^{4}\mathrm{He}

trimer and tetramer [Phys. Rev. A 85, 022502 (2012); Phys. Rev. A 85, 062505 (2012)]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length

{r}_{\mathrm{vdW}}

, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius

{R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}

, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius

{R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}

makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the

^{4}\mathrm{He}

trimer with the realistic potentials.

DOI: 10.1103/physreva.90.052514

Related papers

Linear correlations between 4 He trimer and tetramer energies calculated with various realistic 4 He potentials

Emiko Hiyama, M. Kamimura

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 ${}^{4}$ He 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 ${}^{4}$ He- ${}^{4}$ He 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, $+9.37\phantom{\rule{0.16em}{0ex}}(+3.48)$ mK, $\ensuremath{-}1.20\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}0.46)$ mK, and $+0.16\phantom{\rule{0.16em}{0ex}}(+0.07)$ mK. Further including other realistic ${}^{4}$ He potentials, we calculated the binding energies of the trimer and tetramer ground and excited states, ${B}_{3}^{(0)},\phantom{\rule{0.16em}{0ex}}{B}_{3}^{(1)},\phantom{\rule{0.16em}{0ex}}{B}_{4}^{(0)}$ , and ${B}_{4}^{(1)}$ , 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 ${}^{4}$ He atoms (namely, six types of the generalized Tjon lines are observed). The dimerlike-pair model for ${}^{4}$ He clusters, proposed in the previous work, predicts a simple interaction-independent relation $\frac{{B}_{4}^{(1)}}{{B}_{2}}=\frac{{B}_{3}^{(0)}}{{B}_{2}}+\frac{2}{3}$ , which precisely explains the correlation between the tetramer excited-state energy and the trimer ground-state energy, with ${B}_{2}$ being the dimer binding energy.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 ${}^{4}$He 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 ${}^{4}$He-${}^{4}$He 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, $+9.37\phantom{\rule{0.16em}{0ex}}(+3.48)$ mK, $\ensuremath{-}1.20\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}0.46)$ mK, and $+0.16\phantom{\rule{0.16em}{0ex}}(+0.07)$ mK. Further including other realistic ${}^{4}$He potentials, we calculated the binding energies of the trimer and tetramer ground and excited states, ${B}_{3}^{(0)},\phantom{\rule{0.16em}{0ex}}{B}_{3}^{(1)},\phantom{\rule{0.16em}{0ex}}{B}_{4}^{(0)}$, and ${B}_{4}^{(1)}$, 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 ${}^{4}$He atoms (namely, six types of the generalized Tjon lines are observed). The dimerlike-pair model for ${}^{4}$He clusters, proposed in the previous work, predicts a simple interaction-independent relation $\frac{{B}_{4}^{(1)}}{{B}_{2}}=\frac{{B}_{3}^{(0)}}{{B}_{2}}+\frac{2}{3}$, which precisely explains the correlation between the tetramer excited-state energy and the trimer ground-state energy, with ${B}_{2}$ being the dimer binding energy.

arXiv:1203.3130Physical Review AJun 6, 201252 citations

Universality in Systems of up to Four Particles — From Nuclei to Ultracold Atoms

Christiane H. Schmickler

Universal physics manifests in low-energy phenomena over a large array of different systems from nuclear to atomic physics. Here, two interesting aspects of universality are studied using the Gaussian Expansion Method (GEM). This is a variational method that uses products of Gaussians as basis functions. Gaussian potentials are used as a short-range potential. The first aspect is the behaviour close to the dimer + atom breakup threshold of Efimov states and associated universal tetramers in ultra-cold mixtures of alkali atoms. These can be treated as bosons here. It is shown that trimer and tetramer vanish into the threshold at almost the same point. The predictions of effective Efimov states in the vicinity of the dimer + atom breakup threshold are addressed, but the results are inconclusive. The second aspect is the interplay between universal states of up to four bosons and the Coulomb interaction. This is interesting because it opens up nuclear physics to the investigation. First, the effect of the Coulomb interaction on universal states is studied in natural units. This introduces a scale for the Coulomb potential's strength relative to the strength of the short-range Gaussian potential. A generalised Efimov plot of the binding energies of states of charged bosons versus the Coulomb-modified scattering length is shown. This plot illustrates the impact of different relative strengths of the Coulomb potential on universal states. To complement this analysis, the structure is also calculated via root mean square (rms) radius calculations and contour plots. The results are then applied to the excited state of 17 F, which has a proton halo. The binding energy of this state can be reproduced after fixing the effective range of the potential to the physical value. The N α system proved more problematic. It was found that the ground state of 12 C was too deep to be described within the framework used and the highest excited state of 16 O below the 4α breakup threshold proved difficult to describe. I discuss possible reasons for this. In the last part I studied the zero-range limit in the presence of the Coulomb interaction. I show that my results for the dimer and trimer can be rescaled to coincide with the zero-range result. An extrapolation towards the zero-range limit for the tetramer ground state is also presented.Universal physics manifests in low-energy phenomena over a large array of different systems from nuclear to atomic physics. Here, two interesting aspects of universality are studied using the Gaussian Expansion Method (GEM). This is a variational method that uses products of Gaussians as basis functions. Gaussian potentials are used as a short-range potential. The first aspect is the behaviour close to the dimer + atom breakup threshold of Efimov states and associated universal tetramers in ultra-cold mixtures of alkali atoms. These can be treated as bosons here. It is shown that trimer and tetramer vanish into the threshold at almost the same point. The predictions of effective Efimov states in the vicinity of the dimer + atom breakup threshold are addressed, but the results are inconclusive. The second aspect is the interplay between universal states of up to four bosons and the Coulomb interaction. This is interesting because it opens up nuclear physics to the investigation. First, the effect of the Coulomb interaction on universal states is studied in natural units. This introduces a scale for the Coulomb potential's strength relative to the strength of the short-range Gaussian potential. A generalised Efimov plot of the binding energies of states of charged bosons versus the Coulomb-modified scattering length is shown. This plot illustrates the impact of different relative strengths of the Coulomb potential on universal states. To complement this analysis, the structure is also calculated via root mean square (rms) radius calculations and contour plots. The results are then applied to the excited state of 17 F, which has a proton halo. The binding energy of this state can be reproduced after fixing the effective range of the potential to the physical value. The N α system proved more problematic. It was found that the ground state of 12 C was too deep to be described within the framework used and the highest excited state of 16 O below the 4α breakup threshold proved difficult to describe. I discuss possible reasons for this. In the last part I studied the zero-range limit in the presence of the Coulomb interaction. I show that my results for the dimer and trimer can be rescaled to coincide with the zero-range result. An extrapolation towards the zero-range limit for the tetramer ground state is also presented.

Jan 1, 20200 citations

Zero-energy scattering and bound states in the He4 trimer and tetramer

S. Nakaichi-Maeda, T. K. Lim

Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the $^{4}\mathrm{He}$ atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the $s$ -wave two-body $t$ matrix to a separable form through the unitary-pole approximation and did the same for the $s$ -wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the $^{4}\mathrm{He}$ atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the $s$-wave two-body $t$ matrix to a separable form through the unitary-pole approximation and did the same for the $s$-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.

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M. R. Hadizadeh, Marcelo Takeshi Yamashita, Lauro Tomio +2 more

We evidence the existence of a universal correlation between the binding energies of successive four-boson bound states (tetramers), for large two-body scattering lengths (a), related to an additional scale not constrained by three-body Efimov physics. Relevant to ultracold atom experiments, the atom-trimer relaxation peaks for |a|→∞ when the ratio between the tetramer and trimer energies is ≃4.6 and a new tetramer is formed. The new scale is also revealed for a < 0 by the prediction of a correlation between the positions of two successive peaks in the four-atom recombination process.We evidence the existence of a universal correlation between the binding energies of successive four-boson bound states (tetramers), for large two-body scattering lengths (a), related to an additional scale not constrained by three-body Efimov physics. Relevant to ultracold atom experiments, the atom-trimer relaxation peaks for |a|→∞ when the ratio between the tetramer and trimer energies is ≃4.6 and a new tetramer is formed. The new scale is also revealed for a < 0 by the prediction of a correlation between the positions of two successive peaks in the four-atom recombination process.

arXiv:1101.0378Physical Review LettersSep 23, 201179 citations

Prediction of a weakly bound excited state of Efimov character in a Li7He24 system

M.-S. Wu, Hui-Li Han, Chengbin Li +1 more

We carry out calculations on the van der Waals trimer $^{7}\mathrm{Li}^{4}\mathrm{He}_{2}$ using the mapping method within the frame of hyperspherical coordinates, which allows us to give accurate binding energies and wave functions for both the ground and excited state of the system. When the realistic two-body potentials are adopted, the system presents an excited state which shows Efimov character. We study the range of the interaction strength in which the excited state could exist and find that the state persists within the experiment error band for binding energy of LiHe molecule. We also study the three-body parameter (3BP) of $^{7}\mathrm{Li}^{4}\mathrm{He}_{2}$ system and its relationship with the background scattering length ${a}_{\text{HeHe}}$ . Our calculations demonstrate that the 3BP of $^{7}\mathrm{Li}^{4}\mathrm{He}_{2}$ system is dependent on the value of the scattering length ${a}_{\text{HeHe}}$ , independent of the short-range details of the He-He interaction. The results confirm the prediction of Wang et al. [Phys. Rev. Lett. 109, 243201 (2012)] that the 3BP for a heteronuclear atomic system is universally determined from the van der Waals lengths and the homonuclear scattering length.We carry out calculations on the van der Waals trimer $^{7}\mathrm{Li}^{4}\mathrm{He}_{2}$ using the mapping method within the frame of hyperspherical coordinates, which allows us to give accurate binding energies and wave functions for both the ground and excited state of the system. When the realistic two-body potentials are adopted, the system presents an excited state which shows Efimov character. We study the range of the interaction strength in which the excited state could exist and find that the state persists within the experiment error band for binding energy of LiHe molecule. We also study the three-body parameter (3BP) of $^{7}\mathrm{Li}^{4}\mathrm{He}_{2}$ system and its relationship with the background scattering length ${a}_{\text{HeHe}}$. Our calculations demonstrate that the 3BP of $^{7}\mathrm{Li}^{4}\mathrm{He}_{2}$ system is dependent on the value of the scattering length ${a}_{\text{HeHe}}$, independent of the short-range details of the He-He interaction. The results confirm the prediction of Wang et al. [Phys. Rev. Lett. 109, 243201 (2012)] that the 3BP for a heteronuclear atomic system is universally determined from the van der Waals lengths and the homonuclear scattering length.

Physical Review ADec 8, 201416 citations

Universality and the three-body parameter of 4 He trimers

Pascal Naidon, Emiko Hiyama, Masahito Ueda

We consider a system of three ${}^{4}$ He atoms, which is so far the simplest realistic three-body system exhibiting the Efimov effect, in order to analyze deviations from the universal Efimov three-body spectrum. We first calculate the bound states using a realistic two-body potential, and then analyze how they can be reproduced by simple effective models beyond Efimov's universal theory. We find that the nonuniversal variations of the ground and first excited states can be well reproduced by models parametrized with only three quantities: the scattering length and effective range of the original potential, and a small three-body force. Furthermore, the three-body parameter, which fixes the origin of the infinite set of three-body levels, is found to be consistent with recent experimental observations in other atomic species.We consider a system of three ${}^{4}$He atoms, which is so far the simplest realistic three-body system exhibiting the Efimov effect, in order to analyze deviations from the universal Efimov three-body spectrum. We first calculate the bound states using a realistic two-body potential, and then analyze how they can be reproduced by simple effective models beyond Efimov's universal theory. We find that the nonuniversal variations of the ground and first excited states can be well reproduced by models parametrized with only three quantities: the scattering length and effective range of the original potential, and a small three-body force. Furthermore, the three-body parameter, which fixes the origin of the infinite set of three-body levels, is found to be consistent with recent experimental observations in other atomic species.

arXiv:1109.5807Physical Review AJul 9, 201243 citations

We calculated, using seven realistic

^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}

potentials in the literature, the Efimov spectra of the

^{4}\mathrm{He}

^{4}\mathrm{He}

{r}_{\mathrm{vdW}}

, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius

{R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}

{R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}

makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the

^{4}\mathrm{He}

trimer with the realistic potentials.

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}

He 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}

He-

{}^{4}

He 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\phantom{\rule{0.16em}{0ex}}(+3.48)

mK, $\ensuremath{-}1.20\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}0.46)$ mK, and

+0.16\phantom{\rule{0.16em}{0ex}}(+0.07)

mK. Further including other realistic

{}^{4}

He potentials, we calculated the binding energies of the trimer and tetramer ground and excited states,

{B}_{3}^{(0)},\phantom{\rule{0.16em}{0ex}}{B}_{3}^{(1)},\phantom{\rule{0.16em}{0ex}}{B}_{4}^{(0)}

, and

{B}_{4}^{(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}

He atoms (namely, six types of the generalized Tjon lines are observed). The dimerlike-pair model for

{}^{4}

He clusters, proposed in the previous work, predicts a simple interaction-independent relation

\frac{{B}_{4}^{(1)}}{{B}_{2}}=\frac{{B}_{3}^{(0)}}{{B}_{2}}+\frac{2}{3}

, which precisely explains the correlation between the tetramer excited-state energy and the trimer ground-state energy, with

{B}_{2}

being the dimer binding energy.

Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the

^{4}\mathrm{He}

atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the

s

-wave two-body

t

matrix to a separable form through the unitary-pole approximation and did the same for the

s

-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.

We carry out calculations on the van der Waals trimer

^{7}\mathrm{Li}^{4}\mathrm{He}_{2}

using the mapping method within the frame of hyperspherical coordinates, which allows us to give accurate binding energies and wave functions for both the ground and excited state of the system. When the realistic two-body potentials are adopted, the system presents an excited state which shows Efimov character. We study the range of the interaction strength in which the excited state could exist and find that the state persists within the experiment error band for binding energy of LiHe molecule. We also study the three-body parameter (3BP) of

^{7}\mathrm{Li}^{4}\mathrm{He}_{2}

system and its relationship with the background scattering length

{a}_{\text{HeHe}}

. Our calculations demonstrate that the 3BP of

^{7}\mathrm{Li}^{4}\mathrm{He}_{2}

system is dependent on the value of the scattering length

{a}_{\text{HeHe}}

, independent of the short-range details of the He-He interaction. The results confirm the prediction of Wang et al. [Phys. Rev. Lett. 109, 243201 (2012)] that the 3BP for a heteronuclear atomic system is universally determined from the van der Waals lengths and the homonuclear scattering length.

We consider a system of three

{}^{4}

He atoms, which is so far the simplest realistic three-body system exhibiting the Efimov effect, in order to analyze deviations from the universal Efimov three-body spectrum. We first calculate the bound states using a realistic two-body potential, and then analyze how they can be reproduced by simple effective models beyond Efimov's universal theory. We find that the nonuniversal variations of the ground and first excited states can be well reproduced by models parametrized with only three quantities: the scattering length and effective range of the original potential, and a small three-body force. Furthermore, the three-body parameter, which fixes the origin of the infinite set of three-body levels, is found to be consistent with recent experimental observations in other atomic species.