Universality in Efimov-associated tetramers in ${}^4\mathrm{He}$

We calculated, using seven realistic ${}^4\mathrm{He}\text{−}{}^4\mathrm{He}$ 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 Schrödinger 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 $R_4\approx 3r_{\mathrm{vdW}}$, 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 $R_3\approx 2r_{\mathrm{vdW}}$ 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.

Figure 1

The LM2M2 potential $V_2\left(r\right)$, one of the realistic ${}^4\mathrm{He}$ potentials, is shown as the black solid curve. The dashed blue curve shows the density of the dimer $|{\Psi }_2{\left(r\right)|}^2$ (in arbitrary unit) at $E_2=-1.309$ mK, so weakly bound compared with the potential pocket of $-11$ K. $r_{\mathrm{vdW}}=5.08a_0,a_0$ being the Bohr radius.

Figure 2

Difference in the realistic ${}^4\mathrm{He}$ potentials $V_2\left(r\right)$ between the three example potentials. The dimer energy is $E_2=-1.309$ mK(LM2M2), $-1.615$ mK(PCKLJS), and $-1.744$ mK(SAPT96). The repulsive core in the potentials is $\approx 1000$ K at $r=0.7r_{\mathrm{vdW}}$. The dashed blue curve is the same density of the dimer in Fig. 1 (in arbitrary unit).

Figure 3

Efimov spectrum for the ${}^4\mathrm{He}$ trimer calculated with the seven realistic ${}^4\mathrm{He}$ potentials, LM2M2, TTY, HFD-B, HFD-B3-FCI1, SAPT96, CCSAPT07, and PCKLJS. Trimer energy $E_3^{\left(v\right)}$ as a function of the scattering length $a$ for the ground $(v=0)$ and excited $(v=1)$ states are shown as the solid curves, while the dimer energy is shown as the dashed curve. The curves for different potentials overlap with each other within the line thickness. To clarify the figure, $a$ and $E$ are scaled with the $r_{\mathrm{vdW}}$ and $E_{\mathrm{vdW}}$, and raised to the power $-1/2$ and 1/4, respectively. $a_{-}^{\left(0\right)}$ and $a_{-}^{\left(1\right)}$ denote, respectively, the scattering lengths where the trimer energies $E_3^{\left(0\right)}$ and $E_3^{\left(1\right)}$ intersect the three-atom threshold. The dotted line indicates the scattering length corresponding to the unscaled potential at $\lambda =1$.

Figure 4

Probability density $|f_3^{\left(v\right)}\left(R_3\right){|}^2(v=0,1)$ of the ${}^4\mathrm{He}$ trimer ground (solid curve) and excited (dotted red curve) states versus the three-body hyperradius $R_3$ for $R_3<7r_{\mathrm{vdW}}$. The dotted curve has been multiplied by a factor 463 to emphasize that the two states exhibit the same shape of the strong short-range correlations for $R_3\lesssim 3r_{\mathrm{vdW}}$.

Figure 5

The same as Fig. 4 for $R_3<25r_{\mathrm{vdW}}$ in the log sale, but no factor is multiplied to the dotted curve.

Figure 6

The three-body potential $U_3^{\left(v\right)}\left(R_3\right)(v=0,1)$ defined by Eq. (2.6) for the trimer ground (solid curve) and excited (dotted red curve) states. The two curves overlap with each other within the thickness of the lines. The dash-dotted curve shows the Efimov attraction, Eq. (2.7), and the dashed blue curve is for the density $|f_3^{\left(0\right)}\left(R_3\right){|}^2$ in arbitrary unit.

Figure 7

Efimov spectrum for the ${}^4\mathrm{He}$ tetramer calculated with the seven realistic ${}^4\mathrm{He}$ potentials, LM2M2, TTY, HFD-B, HFD-B3-FCI1, SAPT96, CCSAPT07, and PCKLJS: The scaled tetramer energy $E_4^{\left(v\right)}/E_{\mathrm{vdW}}$ as a function of the scaled-inverse scattering length ${(a/r_{\mathrm{vdW}})}^{-1}$ for the ground $(v=0)$ and excited $(v=1)$ states. The curves for different potentials overlap with each other within the line thickness. The thin solid blue curves denote the trimer spectrum which is the same as that in Fig. 3. $a_{-}^{(4,0)}$ and $a_{-}^{(4,1)}$ denote the critical scattering lengths where the tetramer energies $E_4^{\left(0\right)}$ and $E_4^{\left(1\right)}$ cross the four-atom threshold, respectively.

Figure 8

Probability density $|f_4^{\left(v\right)}\left(R_4\right){|}^2(v=0,1)$ of the ${}^4\mathrm{He}$ tetramer ground (solid curve) and excited (dotted red curve) states as a function of the hyperradius $R_4$ for $R_4<10r_{\mathrm{vdW}}$. The density is normalized as ${\int }_0^{\infty }{\left|f_4^{\left(v\right)}\left(R_4\right)\right|}^{2}d{R}_4=1$. The dotted curve has been multiplied by a factor 25 to emphasize that the two tetramer states exhibit the same shape of the strong short-range correlations for $R_4\lesssim 3.5r_{\mathrm{vdW}}$.

Figure 9

The same as Fig. 8 for $R_4<25r_{\mathrm{vdW}}$ in the log scale, but no factor is multiplied to the dotted curve.

Figure 10

The four-body potential $U_4^{\left(v\right)}\left(R_4\right)(v=0,1)$ defined by Eq. (2.6) for the ${}^4\mathrm{He}$ tetramer ground (solid curve) and excited (dotted red curve) states. The two curves overlap with each other within the thickness of the lines. The dashed blue curve is for the density $|f_4^{\left(0\right)}\left(R_4\right){|}^2$ in arbitrary unit.