Temperature dependence of $p$-wave contacts in a harmonically trapped Fermi gas

TL;DR

The study reveals that the $p$-wave contacts $C_{v,m}$ in a harmonically trapped $^6$Li Fermi gas depend on both the Fermi temperature $T_F$ and the reduced temperature $T/T_F$ due to dipolar splitting and the effective range. By resolving the $m=0$ and $m=\pm1$ channels through anisotropic dissociation of closed-channel molecules, the authors extract $C_{v,m}$ for all three $m$ components and observe near-resonant scaling with $\sqrt{T_F}$, indicating a dominant role for the normalized effective range $k_F R_{\rm e}$. The temperature dependence shows that peak contacts near resonance grow as $T/T_F$ decreases, in line with second-order virial expansion estimates, providing benchmarks for the thermodynamics of resonantly enhanced $p$-wave Fermi gases. Altogether, the work links experimental access to $m$-resolved $p$-wave contacts with virial-theory predictions, advancing understanding of the equation of state and many-body thermodynamics in $p$-wave systems.

Abstract

We study the dependence of the $p$-wave contact on the Fermi temperature $T_F$ and reduced temperature $T/T_F$ based on the number of closed-channel molecules. From the anisotropic pattern of dissociated molecules, we resolve the narrow $m=0$ and $m=\pm1$ dipolar splitting of the $p$-wave Feshbach resonance in $^6$Li, enabling the independent determination of the contact for all three $m$ components. For each component, we identify a near-resonant scaling with $\sqrt{T_F}$, indicating the contribution of the normalized effective range $k_F R_{\rm{e}}$. In addition, we show that the peak contacts observed near resonance increase as $T/T_F$ is lowered, a trend that is accurately captured by estimates based on the second-order virial expansion. Our results, together with estimates from the $p$-wave virial expansion, provide a route toward a complete understanding of the thermodynamics of resonantly enhanced $p$-wave Fermi gases.

Figures (5)

• Figure 1: Observation of the dipolar splitting. (a, b) Momentum distributions of dissociated $p$-wave closed-channel molecules in the $x-z$ plane at various magnetic fields: (a) experimental data averaged over 3–6 shots and (b) numerical calculation. The central peaks in (a) originate from open-channel atoms transferred to state $\ket{2}$. (c) Normalized angular probability distribution $\tilde{P}(\theta)$ of dissociated molecules. Fits to Eq. \ref{['theta_plot']} (solid lines) extract the anisotropy parameter $2\gamma$ being 0.99(2), 0.64(1), and 0.35(1) for $\delta B = 8$, 19, and 33 mG, respectively. (d) Fraction of closed-channel molecules $f_{c,m}$ and normalized contacts $C_{v,m} k_F/N$ as functions of magnetic field detuning $\delta B$ or normalized dimer energy for each $m$ component. Error bars represent the standard deviations of three repetitions.
• Figure 2: Fermi temperature dependence. (a) Anisotropy parameter $\gamma$ versus dimer energy $E_{\mathrm{d},m=\pm1}/E_F$ at various Fermi temperature $T_F$, and the inverse normalized splitting energy $E_F/\delta E$ (right axis). (b) $C_{v,m}$ versus $\sqrt{T_F}$ and the effective range $k_F R_{\mathrm{e}}$ at $E_{\mathrm{d},m}/E_F = 0.7$ (green) and $1.5$ (orange). The shaded area indicates a systematic uncertainty arising from atom losses, shown for $m=0$ and expected to be similar for $m=\pm1$. The solid line is a guide to the eye.
• Figure 3: $p$-wave contact $C_{v,m=+1}$ versus dimer energy $E_{\mathrm{d},m=+1}/E_F$ for various $T/T_F$ at fixed $T_F = 6.8~\mu\mathrm{K}$ ($k_F R=7.6\times10^{-3}$). Solid lines show second-virial estimates. Results for $m=0,-1$ are omitted as they agree with $m=+1$. The shaded bands show systematic uncertainties.
• Figure 4: Peak contact values for $m=0, +1, -1$ at $E_{\rm{d}}/E_F=0.7$, normalized by $k_F R_{\rm{e}}$ at various $T_F$, plotted as a function of $T/T_F$. Temperatures are measured before the interaction quench. The black dashed line shows the second-virial estimate with a shaded systematic uncertainty band.
• Figure S1: Schematic of the measurement of the anisotropic distribution of dissociated $p$-wave molecules using absorption imaging perpendicular to the magnetic field. The dissociated molecules expand on a sphere with radial distribution $n(R)$ and angular dependence $|Y_\ell^m(\Theta,\Phi)|^2$ for $\ell=1$. When imaged along the $y$ axis, the distributions projected onto the $xz$ plane are $z$-weighted for $m=0$ and $x$-weighted for $m=\pm1$.