Probing an effective-range-induced super fermionic Tonks-Girardeau gas with ultracold atoms in one-dimensional harmonic traps

We theoretically investigate an ultracold spin-polarized atomic Fermi gas with resonant odd-channel $(p$-wave) interactions trapped in one-dimensional harmonic traps. We solve the Yang-Yang thermodynamic equations based on the exact Bethe ansatz solution, and predict the finite-temperature density profile and breathing mode frequency by using a local density approximation to take into account the harmonic trapping potential. The system features an exotic super fermionic Tonks-Girardeau (super-fTG) phase, due to the large effective range of the interatomic interactions. We explore the parameter space for such a fascinating super-fTG phase at finite temperature and provide smoking-gun signatures of its existence in both breathing mode frequencies and density profiles. Our results suggest that the super-fTG phase can be readily probed at temperatures of about $0.1T_F$, where $T_F$ is the Fermi temperature. These results are to be confronted with future cold-atom experiments with ${}^6\mathrm{Li}$ and ${}^{40}\mathrm{K}$ atoms.

Figure 1

Contour plot of the squared breathing mode frequency ${({\omega }_b/{\omega }_{\mathrm{ho}})}^2$ as functions of the dimensionless interaction parameters ${\gamma }_2$ and ${\gamma }_1{\gamma }_2/\left(4{\pi }^2\right)$ (see the text for their definitions) in the logarithmic scale. The black dashed line is the zero-temperature analytic result Eq. (19) [19], indicating the transition into the super-fTG regime, either from the weakly interacting limit or the strongly interacting fTG limit. We have taken a typical temperature $T=0.1T{}_{\mathrm{F}}$.

Figure 2

Local sound velocity $c/v_{{}_l}\left(n\right)$ as a function of the local density $n/n_{{}_{\mathrm{F}}}$ for three sets of effective ranges ${\xi }_p={10}^{-4}{a}_{\mathrm{ho}}$ (a, d), $0.5a_{\mathrm{ho}}$ (b, e), and $1.58a_{\mathrm{ho}}$ (c, f) at low temperature $T=0.1T{}_{\mathrm{F}}$ (upper panel) and high temperature $T=T{}_{\mathrm{F}}$ (lower panel). In each subplot at the same effective ranges ${\xi }_p$, the interaction parameters $-w_1/a_{\perp }^3$ of the four different curves are indicated, corresponding to the four highlighted points in the curves of the squared breathing mode frequency, as shown in the insets of Fig. 3. Here, $v_{{}_l}\left(n\right)=\pi \hslash n/m$ is the local Fermi velocity and $T_{\mathrm{F}}=N\hslash {\omega }_{\mathrm{ho}}$ is the Fermi temperature.

Figure 3

Density profiles at different effective ranges: (a) ${\xi }_p={10}^{-4}{a}_{\mathrm{ho}}$, (b) $0.5a_{\mathrm{ho}}$, and (c) $1.58a_{\mathrm{ho}}$, at $T=0.1T{}_{\mathrm{F}}$. The inset shows the squared breathing mode frequency as a function of the interaction parameter $-w_1/a_{\perp }^3$. In the subplot, the value of $-w_1/a_{\perp }^3$ for each density distribution shown in the solid blue (dashed orange, dotted yellow, and dot-dashed purple) line is indicated by using the circle (square, diamond, and star) in the inset. Here, $x_{\mathrm{F}}=\sqrt{2N}{a}_{\mathrm{ho}}$ is the radius of an ideal trapped Fermi gas at zero temperature.

Figure 4

The squared breathing mode frequencies ${({\omega }_b/{\omega }_{\mathrm{ho}})}^2$ as a function of the interaction parameter $-w_1/a_{\perp }^3$, at different effective ranges [upper panel, (a)–(c)] or at different temperatures [lower panel, (d)–(e)], as indicated.