Efimov correlations in strongly interacting Bose gases

We compute the virial coefficients, the contact parameters, and the momentum distribution of a strongly interacting three-dimensional Bose gas by means of a virial expansion up to third order in the fugacity, which takes into account three-body correlations exactly. Our results characterize the nondegenerate regime of the interacting Bose gas, where the thermal wavelength is smaller than the interparticle spacing but the scattering length may be arbitrarily large. We observe a rapid variation of the third virial coefficient as the scattering length is tuned across the three-atom and the atom-dimer thresholds. The momentum distribution at unitarity displays a universal high-momentum tail with a log-periodic momentum dependence, which is a direct signature of Efimov physics. We provide a quantitative description of the momentum distribution at high momentum as measured by P. Makotyn et al. [Nat. Phys. 10, 116 (2014)], and our calculations indicate that the lowest trimer state might not be occupied in the experiment. Our results allow for a spectroscopy of Efimov states in the unitary limit.

Figure 1

(a) Integral equation for the three-body scattering matrix, denoted by a gray box. Simple lines denote atom propagators, and double lines are dimer propagators. (b)–(m) Diagrams that contribute to the density and the momentum distribution up to third order in the fugacity.

Figure 2

(a) Second virial coefficient as a function of scattering length. (b) Third virial coefficient at unitarity as a function of the three-body parameter. The dashed red and dotted green lines in (a) and (b) denote the analytical asymptotic results [41, 42]. (c) Third virial coefficient as a function of scattering length for fixed (along the arrow) ${\kappa }_{*}{\lambda }_T=0.5,1,2,3,5$. (d) Bound-state spectrum. Blue solid lines denote trimer branches, and the red dashed line is the dimer bound-state energy. (e) Third virial coefficient at small and positive scattering length $a\ll {\lambda }_T$. Dashed lines denote the asymptotic result based on the atom-dimer Beth-Uhlenbeck formula (4).

Figure 3

(a) Two-body contact as a function of temperature. Three-body contact at unitarity as a function of (b) temperature and (c) the three-body parameter. The lines in (a) and (b) correspond to (from bottom to top) ${\kappa }_{*}/k_n=1,2,3,4$, and 5. The green lines in (a) and (b) denote the asymptotic high-temperature results. The red and green lines in (c) denote the asymptotic results as stated in the main text.

Figure 4

Third-order contribution $n_3\left(\zeta \right)$ to the momentum distribution at unitarity with ${\kappa }_{*}{\lambda }_T=3$ as a function of dimensionless momentum $\zeta =q{\lambda }_T$. Points denote numerical results; the solid blue line is a guide to the eye. The green and dot-dashed orange lines denote the asymptotic Tan relations (7) obtained with $e^{-\beta E_T}{c}_{2,3}=825.2$ and $e^{-\beta E_T}{c}_3=48.1$.

Figure 5

Full momentum distribution $n\left(\zeta \right)$ up to third order in the fugacity with ${\kappa }_{*}{\lambda }_T=3$ for three different scattering lengths ${\lambda }_T/a=-3.16,0$, and 3.16. Fugacities have been chosen such that $z^3{e}^{\beta E}=0.6$. (a) ${\lambda }_T/a=-3.16$, $\beta E=0$. (b) ${\lambda }_T/a=0$, $\beta E=1.43$. (c) ${\lambda }_T/a=3.16$, $\beta E=5.84$. The notation is the same as in Fig. 4.

Figure 6

Full momentum distribution $n\left(\zeta \right)$ up to third order in the fugacity at unitarity with ${\kappa }_{*}{\lambda }_T=1,3$, and 5. Fugacities have been chosen such that $z^3{e}^{\beta E}=0.6$. (a) ${\kappa }_{*}{\lambda }_T=1$, $\beta E=0.16$. (b) ${\kappa }_{*}{\lambda }_T=3$, $\beta E=1.43$. (c) ${\kappa }_{*}{\lambda }_T=5$, $\beta E=3.98$. The notation is the same as in Fig. 4.

Figure 7

Momentum distribution of spin-up particles in a population-balanced two-component Fermi gas as a function of dimensionless momentum $\zeta =q{\lambda }_T$ including terms up to third order in the fugacity $z$. The scattering lengths are ${\lambda }_T/a=-1$ (green), ${\lambda }_T/a=0$ (red), and ${\lambda }_T/a=1$ (blue). Fugacities were chosen such that $z^2{e}^{\beta E}=0.15$, where $E=E_D$ for $a>0$ and $E=0$ otherwise. Dot-dashed lines denote the corresponding values of the contact as obtained from an independent calculation for the virial coefficients.

Figure 8

Momentum distribution of a unitary Bose gas. $\kappa =q/k_n$ denotes the dimensionless momentum. The solid red and blue lines are the result of the trap-averaged virial expansion with fugacities $z_1=0.5$ and $z_2=0.4$, respectively, corresponding to the experimental data [31] at $\langle n_1\rangle =5.5\times {10}^{12}{\mathrm{cm}}^{-3}$ (orange line) and $\langle n_2\rangle =1.6\times {10}^{12}{\mathrm{cm}}^{-3}$ (green line). The dashed red and blue lines are the corresponding $\mathit{O}\left(z^2\right)$ results. All momentum distributions are normalized to unity $\int d^3\kappa n\left(\kappa \right)/{\left(2\pi \right)}^3=1$.