Single-particle momentum distribution of an Efimov trimer

Experimental progress in the study of strongly interacting ultracold atoms has recently allowed the observation of Efimov trimers. We study theoretically a nonconventional observable for these trimer states, which may be accessed experimentally, the momentum distribution $n(\mathbf{k})$ of the constitutive bosonic particles. The large momentum part of the distribution is particularly intriguing: In addition to the expected $1/k^4$ tail associated with contact interactions, it exhibits a subleading tail $1/k^5$ which is a hallmark of Efimov physics and leads to a breakdown of a previously proposed expression of the energy as a functional of the momentum distribution.

Figure 1

For a free space Efimov trimer at rest composed of three bosonic particles of mass $m$ interacting via a zero-range, infinite scattering length potential, single-particle momentum distribution $n(\mathbf{k})$ as a function of $k$. (a) Numerical calculation from the expression for $n(\mathbf{k})$ appearing in Eq. (18). (b) Numerical calculation (solid line) and asymptotic behavior (26) (dashed line), with the horizontal axis in log scale. The unit of momentum is ${\kappa }_0$, such that the trimer energy is $-\hslash {}^2{\kappa }_0^2/m$.

Figure 2

Cutoff-dependent energy $E(\Lambda )$ as defined in Eq. (32) for a free space infinite scattering length Efimov trimer with a zero-range interaction, as a function of the logarithm of the momentum cutoff $\Lambda$. Solid line: Numerical result obtained via a calculation of the momentum distribution $n(\mathbf{k})$. Symbols $+$: Direct numerical calculation of $E(\Lambda )$ as exposed in Appendix . Dashed sinusoidal line: Asymptotic oscillatory behavior of $E(\Lambda )$ for large $\Lambda$, obtained by omitting $O(1/\Lambda )$ in (35). Dashed horizontal line: Mean value $\mathrm{Ē}$ around which $E(\Lambda )$ oscillates at large $\Lambda$. The values of $\mathrm{Ē}$ obtained analytically (38) and numerically (37) are indistinguishable at the scale of the figure and clearly deviate from the dotted line giving the true energy $E_{\mathrm{trim}}$ of the trimer, exemplifying the failure of an at a first sight convincing application of an energy formula for bosons in three dimensions. The unit of momentum ${\kappa }_0$ is such that the true trimer energy is $E_{\mathrm{trim}}=-\hslash {}^2{\kappa }_0^2/m$.