Observation of Dynamical Super-Efimovian Expansion in a Unitary Fermi Gas

We report an observation of a dynamical super Efimovian expansion in a strongly interacting Fermi gas by engineering time dependent external harmonic trap frequencies. When the trap frequency is tailored as $[1/4t^2+1/t^2\lambda {\log }^2(t/t_{*}){]}^{1/2}$, where $t_{*}$ and $\lambda$ are two controllable parameters, and the change is faster than a critical value, the expansion of such a quantum gas shows novel dynamics that share the same characteristics as the super Efimov effect. A clear double-log periodicity with discrete geometric scaling emerges for the cloud size in the expansion. The universality of such scaling dynamics is verified both in the noninteracting and in the unitarity limit of Fermi gas. Moreover, the measured energy scaling reveals that the potential and internal energy also show double-log periodicity with a $\pi /2$ phase difference, but the total energy is monotonically decreased. Observing super Efimovian evolution represents a paradigm in probing universal properties and allows us in a new way to study many-body nonequilibrium dynamics with experiments.

Figure 1

The schematic of experimental setup (a) and the super Efimovian expansion (b). A scale invariant ultracold gas is held in a harmonic trap consisting of the two dipole-trap laser beams with frequency ${\omega }_0$. Then, with an initial time $t_0$, the trap frequency starts to decrease as $\omega (t)=[1/4t^2+1/t^2\lambda {\log }^2(t/t_{*}){]}^{1/2}$, which $\lambda$ and $t_{*}$ are two control parameters. L1–L3, achromatic lenses; CCD, charge coupled device; AOM; acousto-optic modulator.

Figure 2

The expansion dynamics of the cloud. (a),(b) and (c),(d) are the mean axial cloud size ${\sigma }_z$ versus the expansion time $t_f$ and the dimensionless axial mean square cloud size ${\sigma }_z^2/[\tau \log (\tau )]$ versus the dimensionless time $\log [\log (\tau )]$ at $B=832$ Gauss; unitary Fermi gas ($B=528$ Gauss; an ideal noninteracting Fermi gas), respectively, where $\tau =(t+t_0)/t_{*}$. Black, blue, and green dots are measured data with $t_0=4\mathrm{ms}$ (${\lambda }_z=2.7\times {10}^{-3}$), $t_0=5\mathrm{ms}$ (${\lambda }_z=1.5\times {10}^{-3}$), and $t_0=6\mathrm{ms}$ (${\lambda }_z=9\times {10}^{-4}$), respectively. Red dots are the expansion size of the cloud when ${\lambda }_z$ is chosen as the critical value ${\lambda }_z=4$. Solid red curves for the theoretical fits based on Eq. (1). Here $t_{*}=0.2\mathrm{ms}$. Error bars represent the standard deviation of the statistic.

Figure 3

The expansion dynamics of the cloud with AR $F=1.3$. (a) is the mean axial(radial) cloud size ${\sigma }_z({\sigma }_x)$ versus time, while (b) is the dimensionless mean square axial(radial) cloud size ${\sigma }_z^2(t)/(\tau \log \tau )[{\sigma }_x^2(t)/(\tau \log \tau )]$ versus the dimensionless time $\log [\log (\tau )]$ at unitary limits. Dots are measured results and the solid lines are best fits. The parameters are ${\omega }_{x0}=2\pi \times 480\mathrm{Hz}$ and ${\omega }_{y0}={\omega }_{z0}=2\pi \times 370\mathrm{Hz}$, $t_0=4\mathrm{ms}$ and $t_{*}=0.1\mathrm{ms}$. Error bars represent the standard deviation of the statistic.

Figure 4

The universality of the super Efimovian expansion. (a) $s_0$ as a function of $t_0$. (b) $s_0$ as a function of $\gamma$, where $\gamma =\sqrt{1/{\lambda }_z-1/4}$. The solid lines are the linear fitting curves and the dashed lines are $s_0={\omega }_b\gamma$ with ${\omega }_b=2$ for the noninteracting fermions (blue curves) and ${\omega }_b=\sqrt{12/5}$ for the unitary Fermi gas (red curves). The other parameters are the same as Fig. 2.

Figure 5

The axial potential (internal) energy ratio versus $t_f$ (a) and the axial energy scaling versus $\log [\log (\tau )]$ (b) for the super Efimovian expansion. The dots are measured data and the solid curves are the theoretical predictions. The blue dots are the scaled potential energy and brown dots are the scaled internal energy. The inset is the dependence of the total energy on the expansion time. Error bars represent the standard deviation of the statistic. The other parameters are the same as Fig. 2.