Crossover in the Efimov spectrum

A filtering method is introduced for solving the zero-range three-boson problem. This scheme permits solving the original Skorniakov Ter-Martirosian integral equation for an arbitrary large ultraviolet cutoff and avoiding the Thomas collapse of the three particles. The method is applied to a more general zero-range model including a finite-background two-body scattering length and the effective range. A crossover in the Efimov spectrum is found in such systems and a specific regime emerges where Efimov states are long-lived.

Figure 1

Functions $k/q\sqrt{1+3k^2/(4q^2)}{\varphi }_n(k/q)$ obtained at unitarity ($\left|a\right|=\infty ,R^{\star }=0$) from Eq. (11) for the first eight trimers (including the ground state). The computation was performed with the UV cutoff ${\Lambda }_3=5\times {10}^2{\kappa }^{\star }$. The function $\sin \{s_0\mathrm{arcsinh}[k\sqrt{3}/(2q)]\}$ appearing in the exact solution, Eq. (7), is superimposed and is not distinguishable from the functions obtained by solving Eq. (11).

Figure 2

Trimer spectrum of Eq. (11) at resonance ($\left|a\right|=\infty$) and for $a_{\mathrm{bg}}<0$. (a) Binding wave numbers $q$ (solid lines) as a function of $a_{\mathrm{bg}}$ for a constant and large value of the width radius $R^{\star }$; dashed vertical lines correspond to the thresholds $a_{\mathrm{bg}}{\kappa }^{\star }\simeq -1.51e^{n\pi /s_0}$, where $n=0,1,2$. This figure illustrate the crossover from a narrow FR ($R^{\star }{\kappa }^{\star }\gg 1$ and $\left|a_{\mathrm{bg}}\right|\ll R^{\star }$) to a shape resonance ($\left|a_{\mathrm{bg}}\right|{\kappa }^{\star }$ is large). (b) Binding wave numbers as a function of the width radius for a constant and large value of $\left|a_{\mathrm{bg}}\right|{\kappa }^{\star }$. This figure illustrates the crossover from a broad FR ($R^{\star }{\kappa }^{\star }\ll 1$) to a narrow FR superimposed on a shape resonance.

Figure 3

Eigenfunctions $k^2{\varphi }_n(k)$ for the deepest trimer states obtained after normalization of Eq. (11) for $a_{\mathrm{bg}}{\kappa }^{\star }=-{10}^2$ and $R^{\star }{\kappa }^{\star }={10}^4$. Solid lines: eigenstates with a binding wave number given approximately by the spectrum of a narrow resonance [$q_p={\kappa }^{\star }\hspace{0.5em}{}^{\prime }\exp (p\pi /s_0)$]. Dashed lines: eigenstates with a binding wave number given approximately by the spectrum of a broad resonance [$q_n={\kappa }^{\star }\exp (n\pi /s_0)$]. Dotted vertical line: position of the node imposed by Eq. (10).

Figure 4

Same as Fig. 3, but for $a_{\mathrm{bg}}{\kappa }^{\star }=-{10}^2$ and $R^{\star }{\kappa }^{\star }={10}^8$. In this regime, characterized by the inequality $R^{\star }\gg \left|a_{\mathrm{bg}}\right|$, there is a clear separation of scale for the two types of eigenstates: the deepest ones of the “broad type” are located in the momentum region $k\gtrsim 1/\left|a_{\mathrm{bg}}\right|$; those of the “narrow type” are located in the momentum region $k\lesssim {\kappa }^{\star }\hspace{0.5em}{}^{\prime }$.

Figure 5

Absolute value of the eigenfunctions $|\varphi (k)|$ in Fig. 4 ($a_{\mathrm{bg}}{\kappa }^{\star }=-{10}^2,R^{\star }{\kappa }^{\star }={10}^8$) on a logarithmic scale. The figure shows that even in this regime, where “broad-type” and “narrow-type” eigenfunctions are well separated, eigenfunctions share the same 0 at $k=k_{\mathrm{reg}}$. Narrow-type eigenstates experience a $k^{-4}$ power law in the region $1\ll k{R}^{\star }\ll \sqrt{\frac{R^{\star }}{\left|a_{\mathrm{bg}}\right|}}$.

Figure 6

Square modulus of the $s$-wave atom-dimer scattering amplitude in the resonant regime ($\left|a\right|=\infty ,a_{\mathrm{bg}}>0$) as a function of the wave number $q=\sqrt{q_{\dim }^2-3k_0^2/4}$, where $k_0$ is the collisional momentum and $q_{\dim }$ is the dimer’s binding wave number. The plot is in units of ${\kappa }^{\star }$. (a) $R^{\star }{\kappa }^{\star }={10}^4$ and $a_{\mathrm{bg}}{\kappa }^{\star }=5\times {10}^2$; dashed vertical lines, “narrow-state” spectrum from Refs. [18] and [19]. (b) Details of the resonant peak at $q=q_0$ in (a): stars, results of the computation; solid line, fit using the Fano profile. (c) $R^{\star }{\kappa }^{\star }={10}^4$ and $a_{\mathrm{bg}}{\kappa }^{\star }=5\times {10}^6$; dashed vertical lines, spectrum of the universal theory [1, 13].