Spectral Response and Contact of the Unitary Fermi Gas

We measure radio frequency (rf) spectra of the homogeneous unitary Fermi gas at temperatures ranging from the Boltzmann regime through quantum degeneracy and across the superfluid transition. For all temperatures, a single spectral peak is observed. Its position smoothly evolves from the bare atomic resonance in the Boltzmann regime to a frequency corresponding to nearly one Fermi energy at the lowest temperatures. At high temperatures, the peak width reflects the scattering rate of the atoms, while at low temperatures, the width is set by the size of fermion pairs. Above the superfluid transition, and approaching the quantum critical regime, the width increases linearly with temperature, indicating non-Fermi-liquid behavior. From the wings of the rf spectra, we obtain the contact, quantifying the strength of short-range pair correlations. We find that the contact rapidly increases as the gas is cooled below the superfluid transition.

Figure 1

(a) Thermal evolution of rf spectra. The Rabi frequency is ${\mathrm{\Omega }}_R=2\pi \times 0.5\mathrm{kHz}$ and the pulse duration is $T_{\text{Pulse}}=1\mathrm{ms}$. The solid lines are guides to the eye. (b) Frequency of the peak ($E_p=-\hslash \omega$) of the rf spectra as a function of temperature shown as white dots on an intensity plot of the rf response. The grey solid line is a solution to the Cooper problem at nonzero temperature [52]. (c) The full width at half maximum $\mathrm{\Gamma }$ of the rf peak as a function of $T/T_F$. The black dotted-dashed line $\mathrm{\Gamma }/E_F=1.2\sqrt{T_F/T}$ shows the temperature dependence of the width due to scattering in the high-temperature gas [32, 60]. The grey triangles are the corresponding width measurements of a highly spin-imbalanced gas [57]. The horizontal black dotted line represents the Fourier broadening of $0.1E_F$ [52]. The vertical dashed red line in both (b) and (c) marks the superfluid transition [14].

Figure 2

Rf spectrum at high frequencies. Here, the temperature of the gas is $T/T_F=0.10(1)$, the pulse duration is $T_{\text{Pulse}}=1\mathrm{ms}$, and the Rabi frequencies are $2\pi \times 536\mathrm{Hz}$ (light blue circles), $2\pi \times 1.20\mathrm{kHz}$ (medium blue triangles), and $2\pi \times 3.04\mathrm{kHz}$ (dark blue squares). The black solid line shows a fit of Eq. (1) to the data, while the grey dashed line shows the fit neglecting final state interactions. The contact can be directly obtained from the transfer rate at a fixed detuning of 60 kHz ($\hslash \omega /E_F\sim 7.1$) (dotted-dashed vertical line). Inset: we vary the pulse time at this fixed detuning, and extract the initial slope (dashed line) of the exponential saturating fit (solid line). The rf transfer rate obtained from the initial linear slope is shown as the red diamond in the main plot. Here, ${\mathrm{\Omega }}_R=2\pi \times 1.18\mathrm{kHz}$.

Figure 3

The dimensionless contact $C/N{k}_F$ (a) and condensate fraction $N_0/N$ (b) of the unitary Fermi gas as a function of the reduced temperature $T/T_F$. Our measurements of the contact (red points) are compared with a number of theoretical estimates: bold-diagrammatic Monte Carlo (BDMC) [38], quantum Monte Carlo (QMC) [37], Luttinger-Ward (LW) [32], large $N$ [10], and Gaussian pair fluctuations (GPF) [36]. Also shown is the homogeneous contact obtained from the equation of state at the École normale supérieure (ENS-EOS) [62], from loss rate measurements (ENS-L) [66], and from rf spectroscopy by the JILA group [18] across a range of temperatures. The vertical blue dotted lines and light blue shaded vertical regions mark $T_c/T_F=0.167(13)$ [14]. The inset of (a) shows the contact over a wider range of temperatures and marks the high-temperature agreement with the third order virial expansion. The error bars account for the statistical uncertainties in the data.