Universality and the three-body parameter of ${}^4$He trimers

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.

Figure 1

LM2M2 potential [33] used for the realistic calculations. The dotted curve indicates the van der Waals asymptote $-C_6/r^6$.

Figure 2

Jacobi coordinates for all of the rearrangement channels ($c=1\sim 3$) of the ${}^4$He trimer system. The wave function of the three ${}^4$He atoms is to be symmetrized.

Figure 3

Ratio between the effective range $r_e$ and scattering length $a$ of the scaled LM2M2 potential as a function of the scaling factor $\lambda$. The vertical solid line indicates the physical value ($\lambda =1$). The dashed vertical line indicates the scaling factor for which the scattering length diverges ($1/a\to 0$). The dashed curve shows the result based on the analytical formula (16) using the values of $\overline{a}$ and $a$ as a function of $\lambda$.

Figure 4

${}^4$He dimer energy $E$ as a function of scattering length $a$. To clarify the figure, these quantities are normalized by $E_e={\hslash }^2/m{r}_e^2$ and $r_e$, and raised to the power $1/4$ and $-1/2$, respectively. The solid red curve corresponds to the dimer of the LM2M2 potential scaled by a varying factor $\lambda$. The physical scattering length of ${}^4$He ($\lambda =1$) is indicated by the vertical gray line. The dotted line represents the universal limit of small binding energy and large scattering length [Eq. (14) with $a\left(E\right)\to a$ ]. The dashed curve represents the two solutions of the effective-range approximation of Eq. (14). The dot-dashed curve corresponds to the results of the separable Gaussian potential (7).

Figure 5

Efimov spectrum for ${}^4$He: trimer energy $E$ as a function of the scattering length $a$. To clarify the figure, these quantities are normalized by $E_e={\hslash }^2/m{r}_e^2$ and $r_e$, and raised to the power $1/4$ and $-1/2$, respectively. The zero of the energy axis corresponds to the three-body threshold. The red solid curves are the energy curves obtained by rescaling the LM2M2 potential. The energy of both trimers and the dimer (rightmost curve) are displayed. The vertical gray line indicates the scattering length corresponding to the unscaled potential, i.e., physical ${}^4$He. The dashed curves correspond to the Efimov spectrum according to the universal theory. The three-body parameter is adjusted to match the first-excited trimer state. The straight dashed line corresponds to the dimer energy in the universal limit of large scattering length.

Figure 6

Square root of the ratio between the energies $E_T^{\left(1\right)}-E_D$ and $E_T^{\left(2\right)}-E_D$ of the first two trimers of different models as a function of the inverse scattering length (in units of $r_e$). Here, the trimer energy is measured from the dimer threshold $E_D$. The solid gray curve represents the ratio in the universal theory. The solid red curve is the result for the scaled helium LM2M2 potential. Other curves represent the results for the universal theory with two-body corrections (green dot-dashed), the local Gaussian potential model with (black dots) and without (black dashes) a three-body interaction, and the separable Gaussian potential model with (blue spaced dots) and without (blue long dashes) a three-body interaction. The vertical line indicates the point where the scattering length is infinite.

Figure 7

Plot similar to Fig. 5. Here again, the ${}^4$He curves are indicated by solid red curves, and results taking into account two-body corrections to the universal theory are indicated by dashed curves.

Figure 8

Three-body cutoff parameter $P$ adjusted to get agreement with the LM2M2 results as a function of energy for the ground-state trimer (dots) and the first excited-state trimer (squares). One can see that their variations are inconsistent.

Figure 9

Plot similar to Fig. 5 for the calculations with the Gaussian potential given by Eq. (6). The dotted curves correspond to two-body interactions only, while the dashed curves correspond to calculations with an additional three-body interaction.

Figure 10

Plot similar to Fig. 5 for the calculations with the separable Gaussian potential given by Eq. (7). The dotted curves correspond to two-body interactions only, while the dashed curves correspond to calculation with an additional three-body interaction. The symbols show the measured dissociation point of the ground-state trimer for different species. One data point from the experiments with potassium lies outside the group of data points, and it is not clear whether this is a special feature of that species breaking the observed tendency or due to a misinterpretation of the data.