Manifestations of the Efimov effect for three identical bosons

We present numerical solutions of the Schrödinger equation for three identical bosons from very large to essentially infinite two-body $s$-wave scattering lengths $a$. We have considered scattering lengths up to $2\times {10}^5\mathrm{a.u.}$ and solved the hyperangular part of the Schrödinger equation for distances up to ${10}^6\mathrm{a.u.}$ From these, we obtained the three-body effective potentials, hyperspherical channel functions and the asymptotic behavior of the nonadiabatic couplings in order to characterize the main aspects of the Efimov states. These results allow us to test and quantify the assumptions related to the Efimov effect.

Figure 1

(Color online) The two-body $s$-wave scattering length $a$ as a function of the potential strength $D$. When the two-body potential is deep enough to support an additional bound state, $a$ passes through a pole, indicated by $D_{\mathit{I}}$, $D_{\mathit{II}}$, and $D_{\mathit{III}}$.

Figure 2

(Color online) The three-body effective potential for (a) $a>0$ and (b) $a<0$, calculated near pole $D=D_{\mathit{I}}$. For $a>0$, the system has only one bound state, while for $a<0$ the system has none.

Figure 3

(Color online) Three-body effective potentials for (a) $a>0$ and (b) $a<0$, with $\mid a\mid =500$, 10 000 100 000, and $200000\mathrm{a.u.}$ (dashed lines). Also shown is the numerical result for $\mid a\mid =\infty$ (solid line). The horizontal dashed line at $-s_0^2$ is the Efimov potential, Eq. (12), and the vertical lines correspond to $R=r_0$ and $\mid a\mid$.

Figure 4

(Color online) The attractive dipole effective potential for different number of two-body bound states. $D_{\mathit{I}}$, $D_{\mathit{II}}$, $D_{\mathit{III}}$ (as given in Fig. 1), and $D_{\mathit{IV}}$ (not shown in Fig. 1) indicate the effective potentials with no $s$-wave bound states, and with one, two, and three $s$-wave bound states, respectively.

Figure 5

(Color online) Three-body effective potentials associated with the continuum channels for $\mid a\mid =\infty$ (solid lines) and for $a=8000\mathrm{a.u.}$ (dashed lines). The dotted lines correspond to the eigenvalues of the kinetic energy; and the vertical dot-dashed lines to the boundaries $r_0$ and $a$.

Figure 6

$J^{\pi }=0^{+}$ channel functions $\Phi (R;\Omega )$ for fixed values of the hyperradius $R$. The first and second lines show the bound channel (BC) and the lowest continuum channel (CC) for $a=500\mathrm{a.u.}$ The third line shows the Efimov channel (EC), demonstrating that it is insensitive to changes in $R$.

Figure 7

(Color online) Channel function for the weakly bound channel at $R=10000\mathrm{a.u.}$ for $a=500$ and $1000\mathrm{a.u.}$ The $\mid a\mid =\infty$ channel function is also shown for comparison.

Figure 8

(Color online) The nonadiabatic couplings between a bound channel and the Efimov channel for finite (dashed lines) and infinite (solid line) scattering lengths. For $a>0$ (a) the Efimov channel is associated with the weakly bound channel; and for $a<0$ (b) with the lowest continuum channel.

Figure 9

(Color online) The nonadiabatic couplings between the Efimov channel and a continuum channel for finite (dashed lines) and infinite (solid line) scattering lengths. For $a>0$ (a) the Efimov channel is associated with the weakly bound channel; and for $a<0$ (b) with the lowest continuum channel.

Figure 10

(Color online) The diagonal nonadiabatic coupling for the Efimov channel for finite (dashed lines) and infinite (solid line) scattering lengths. For $a>0$ (a) the Efimov channel is associated with the weakly bound channel; and for $a<0$ (b) with the lowest continuum channel.