Quantum Anomaly and 2D-3D Crossover in Strongly Interacting Fermi Gases

We present an experimental investigation of collective oscillations in harmonically trapped Fermi gases through the crossover from two to three dimensions. Specifically, we measure the frequency of the radial monopole oscillation or breathing mode in highly oblate gases with tunable interactions. The breathing mode frequency is set by the adiabatic compressibility and probes the thermodynamic equation of state. In 2D, a dynamical scaling symmetry for atoms interacting via a $\delta$ potential predicts the breathing mode to occur at exactly twice the harmonic confinement frequency. However, a renormalized quantum treatment introduces a new length scale which breaks this classical scale invariance resulting in a so-called quantum anomaly. Our measurements deep in the 2D regime lie above the scale-invariant prediction for a range of interaction strengths providing evidence for the quantum anomaly and signifying the breakdown of an elementary $\delta$-potential model of atomic interactions. By varying the atom number we can tune the chemical potential and see the breathing mode frequency evolve smoothly between the 2D to 3D thermodynamic limits.

Figure 1

(a) The radial breathing mode is excited in the $x\text{−}y$ plane by applying a five cycle modulation with a triangular envelope to the magnetic field that provides the radial confinement. The modulation amplitude is chosen such that the cloud radius changes by a few percent after the driving. An in situ absorption image is taken after a variable hold time of up to 250 ms. Inset: Schematic of the breathing mode. (b) Representative data set showing $n_0/{\sigma }^2$ as a function of the hold time for $\ln (k_F^{\mathrm{HO}}{a}_{2\mathrm{D}})=3.3$ with $N/N_{2\mathrm{D}}^{(\mathrm{ideal})}=0.3$, as described in the text. These data yield ${\omega }_B/(2\pi )=46.03\pm 0.35\mathrm{Hz}$.

Figure 2

Relative shift of the breathing mode frequency $\delta {\omega }_B/{\omega }_r\equiv ({\omega }_B-2{\omega }_r)/{\omega }_r$ from the scale-invariant value of $2{\omega }_r$ (black dashed line) as a function of the interaction strength $\ln (k_F^{\mathrm{HO}}{a}_{2\mathrm{D}})$. The temperatures of all clouds used for the above data lie in the range of $T/T_F=0.18\pm 0.04$.

Figure 3

Frequency of the breathing mode through the 2D to 3D crossover for, ${\ell }_z/a_{3\mathrm{D}}=0.53$ (red circles), 0 (blue diamonds), $-0.48$ (purple squares), and $-2.12$ (gray stars). The black dashed line indicates the scale-invariant 2D breathing mode frequency $2{\omega }_r$, red solid line indicates the theoretical breathing mode frequency $\sqrt{3}{\omega }_r$ for a unitary Fermi gas in the 3D thermodynamic limit, and the blue dot-dashed line $\sqrt{10/3}{\omega }_r$ shows the limit for a weakly interacting (thermodynamically 3D) Bose gas.