Few-body states of bosons interacting with a heavy quantum impurity

We consider the problem of a fixed impurity coupled to a small number $N$ of noninteracting bosons. We focus on impurity-boson interactions that are mediated by a closed-channel molecule, as is the case for tuneable interatomic interactions in cold-atom experiments. We show that this two-channel model can be mapped to a boson model with effective boson-boson repulsion, which enables us to solve the three-body $\left(N=2\right)$ problem analytically and determine the trimer energy for impurity-boson scattering lengths $a>0$. By analyzing the atom-dimer scattering amplitude, we find a critical scattering length $a^{*}$ at which the atom-dimer scattering length diverges and the trimer merges into the dimer continuum. We furthermore calculate the tetramer energy exactly for $a>0$ and show that the tetramer also merges with the continuum at $a^{*}$. Indeed, since the critical point $a^{*}$ formally resembles the unitary point $1/a=0$, we find that all higher-body bound states (involving the impurity and $N>1$ bosons) emerge and disappear at both of these points. We show that the behavior at these “multibody resonances” is universal, since it occurs for any model with an effective three-body repulsion involving the impurity. Thus we see that the fixed-impurity problem is strongly affected by a three-body parameter even in the absence of the Efimov effect.

Figure 1

(a),(b) Trimer spectrum of two identical bosons of mass $m$ interacting near resonantly with an impurity atom of mass (a) $M=m$ and (b) $M=\infty$. The spectrum in (a) was calculated in Ref. [14] and shows how trimers can be Borromean when $M$ is finite. The factor of 2 in the vertical axis label in (b) ensures that near the unitarity limit the dimer lines (dotted) are identical to those in (a). We define $a^{*}$ as the scattering length at which the ground-state trimer crosses into the atom-dimer continuum. (c) Three-body parameter $a_{-}$ (the scattering length at which the ground-state trimer crosses into the three-atom continuum) and (d) atom-dimer resonance $a^{*}$ in units of their value at $M=m$, both as a function of mass ratio. The results are obtained within the two-channel model (2) with an effective range $r_0$ (black, solid), or within the $\mathrm{\Lambda }$ model (red, dashed).

Figure 2

Three-body function $Z\left(E\right)$ within the interval $-2E_{\mathrm{B}}<E<-E_{\mathrm{B}}$ for several values of $|r_0|/a$. For $a/|r_0|<a^{*}/\left|r_0\right|\simeq 1/0.31821\cdots ,Z\left(E\right)$ is positive within the plotted energy range, implying that no trimer exists in this range when $U\to \infty$.

Figure 3

Inverse correction $\frac{E_{\text{B}}}{E_3+2E_{\text{B}}}$ as a function of $1/a$ for the two-channel model (green solid) and $\mathrm{\Lambda }$ model (blue dash-dotted). The scattering length $a$ is in units of either $|r_0|$ or $1/\mathrm{\Lambda }$ for the different models. The dashed line is a guide to the eye with slope $1/2\pi$.

Figure 4

Phase shift for the atom-dimer scattering for several values of $|r_0|/a$. When $|r_0|/a=|r_0|/a^{*}\simeq 0.31821\cdots$, the slope at $q=0$ diverges and changes sign.

Figure 5

Scattering length (blue solid) and the effective range (red dashed) of the atom-dimer scattering. The scattering length diverges at $|r_0|/a=0$ (the unitarity limit) and 0.31821 (atom-dimer resonance), while $r_{\mathrm{ad}}$ is finite in the latter case.

Figure 6

Inverse energy correction $\frac{E_{\text{B}}}{E_4+3E_{\text{B}}}$ as a function of $1/a$ calculated within the two-channel model (red solid) and the $\mathrm{\Lambda }$ model (blue dash-dotted). The scattering length $a$ is in units of either $|r_0|$ or $1/\mathrm{\Lambda }$ for the different models. The dashed line is a guide to the eye with slope $1/6\pi$.