Reduced Quantum Anomaly in a Quasi-Two-Dimensional Fermi Superfluid: Significance of the Confinement-Induced Effective Range of Interactions

A two-dimensional (2D) harmonically trapped interacting Fermi gas is anticipated to exhibit a quantum anomaly and possesses a breathing mode at frequencies different from a classical scale-invariant value ${\omega }_B=2{\omega }_{\perp }$, where ${\omega }_{\perp }$ is the trapping frequency. The predicted maximum quantum anomaly ($\sim 10\%$) has not been confirmed in experiments. Here, we theoretically investigate the zero-temperature density equation of state and the breathing mode frequency of an interacting Fermi superfluid at the dimensional crossover from three to two dimensions. We find that the simple model of a 2D Fermi gas with a single $s$-wave scattering length is not adequate to describe the experiments in the 2D limit, as commonly believed. A more complete description of quasi-2D leads to a much weaker quantum anomaly, consistent with the experimental observations. We clarify that the reduced quantum anomaly is due to the significant confinement-induced effective range of interactions.

Figure 1

The column density $n_{2\mathrm{D}}=\int dzn(z)$ of an ideal Fermi gas, in units of $a_z^{-2}=M{\omega }_z/\hslash$, at the dimensional crossover from 2D to 3D. (Inset) Predicted breathing mode frequencies of a 2D interacting Fermi gas at $T=0$ using QMC EOS (dashed line) [8] and GPF EOS (crosses) calculated in this Letter, and at $T=0.42T_F$ using virial expansion (solid line) [18], compared with the experimental data by Vogt et al. (stars, $0.42T_F$) [12], Holten et al. (squares, $0.10-0.18T_F$) [16], and Peppler et al. (circles, $0.14-0.22T_F$) [17].

Figure 2

The density EOS of a unitary Fermi gas at the dimensional crossover, calculated using the (a) mean-field and (b) GPF theories. The anticipated behavior in the 3D limit is shown by the blue dot-dashed lines, and the predictions of the 2D models without and with the effective range of interactions are plotted by the red dashed lines and green asterisks, respectively. The inset in (a) highlights the EOS near the minimum chemical potential ${\mu }_m=(\hslash {\omega }_z-{\epsilon }_B)/2\simeq 0.378\hslash {\omega }_z$. The inset in (b) compares the 2D EOS with contact interactions, predicted by QMC and the GPF theory.

Figure 3

The breathing mode frequency of a unitary Fermi gas at the dimensional crossover, as a function of $N/N_{2\mathrm{D}}$. The squares and circles are the experimental data, measured by Holten et al. [16] at $a_z/a_{3\mathrm{D}}\simeq -0.35$ and Peppler et al. [17] at $a_z/a_{3\mathrm{D}}=0$. The GPF prediction of the two-channel 2D model with the effective range of interactions is also shown by green asterisks (see text for details).

Figure 4

The breathing mode frequency of a strongly interacting 2D Fermi gas at the lowest experimental number of atoms $N/N_{2\mathrm{D}}\sim 0.2$. The squares and circles are the experimental data by Holten et al. ($0.10-0.18T_F$) [16] and Peppler et al. ($0.14-0.22T_F$) [17], respectively. The lines show the predictions of the two-channel 2D model at different effective range of interactions, within the GPF theory.