Expansion of harmonically trapped interacting particles and time dependence of the contact

We study the expansion of an interacting atomic system at zero temperature, following its release from an isotropic three-dimensional harmonic trap and calculate the time dependence of its density and momentum distribution, with special focus on the behavior of the contact parameter. We consider different quantum systems, including the unitary Fermi gas of infinite scattering length, the weakly interacting Bose gas, and two interacting particles with highly asymmetric mass imbalance. In all cases analytic results can be obtained, which show that the initial value of the contact, fixing the $1/k^4$ tail of the momentum distribution, disappears for large expansion times. Our results raise the problem of understanding the recent experiment of R. Chang et al. [Phys. Rev. Lett. 117, 235303 (2016)] carried out on a weakly interacting Bose gas of metastable ${}^4\mathrm{He}$ atoms, where a $1/r^4$ tail in the density distribution was observed after a large expansion time, implying the existence of the $1/k^4$ tail in the asymptotic momentum distribution.

Figure 1

(a) The density distribution $\rho (r,\tau )$ and (b) momentum distribution $n(k,\tau )$ in the presence of a regularized pseudopotential with unitary interaction applied to the two-body problem. Different lines are the distributions at different expansion time $\tau =0$ (solid red), ${10}^2$ (dashed green), and ${10}^4$ (dotted blue), and the asymptotic momentum distribution (dot-dashed black). Results are shown in log-log scale plot. For finite expansion time, the density distribution $\rho (r,\tau )$ exhibits a $1/r^2$ behavior at small $r$ and the momentum distribution $n(k,\tau )$ exhibits a $1/k^4$ behavior at large $k$.

Figure 2

Plot of $\left|c\right(q\left)\right|$ in the absence of interaction (solid blue line), in the presence of interaction (dashed green line), and after quenching the scattering length to infinity (dotted orange line). Results are shown in log-log scale plot. The absolute value is added because $c\left(q\right)$ changes sign at the dips of the blue and orange lines. The explicit $q$ dependence of $c\left(q\right)$ at small and large $q$ is explained in the main text.

Figure 3

Plot of density distribution (a) in the absence and (b) in the presence of interaction in the expansion. Different lines are the distributions at different expansion time $\tau =0$ (solid red), 1 (dashed green), and 5 (dotted blue). Results are shown in log-log scale plot. The density distribution $\rho (r,\tau )$ exhibits a $1/r^4$ behavior at large $r$ if the interaction is turned off, whereas it exhibits a $1/r^{12}$ tail followed by a tail of $1/r^{14}$ if the interaction is present.

Figure 4

Plot of momentum distribution in the presence of interaction in the expansion. Different lines are the distributions at different expansion time $\tau =0$ (solid red), 1 (dashed green), and 5 (dotted blue) and the asymptotic momentum distribution (dot-dashed black). Results are shown in log-log scale plot. At large $k$, the momentum distribution exhibits a tail of $1/k^4$, whereas the asymptotic momentum distribution exhibits a $1/k^{12}$ tail followed by a $1/k^{14}$ tail.

Figure 5

Plot of (a) density and (b) momentum distributions in the expansion with the scattering quenched to $a=\infty$ at $\tau =+0$. Different lines are the distributions at different expansion time $\tau =0$ (solid red), 1 (dashed green), and 5 (dotted blue). The dot-dashed black line indicates the position of the large momentum tail for the in-trap equilibrium system with unitarty interaction. Results are shown in log-log scale plot. At large expansion time, the density distribution exhibits a tail of $1/r^6$ at large $r$, whereas the momentum distribution exhibits a tail of $1/k^4$ at large $k$ which first increases at short time and then decreases.