Excitation Spectrum and Superfluid Gap of an Ultracold Fermi Gas

Ultracold atomic gases are a powerful tool to experimentally study strongly correlated quantum many-body systems. In particular, ultracold Fermi gases with tunable interactions have allowed to realize the famous BEC-BCS crossover from a Bose-Einstein condensate (BEC) of molecules to a Bardeen-Cooper-Schrieffer (BCS) superfluid of weakly bound Cooper pairs. However, large parts of the excitation spectrum of fermionic superfluids in the BEC-BCS crossover are still unexplored. In this work, we use Bragg spectroscopy to measure the full momentum-resolved low-energy excitation spectrum of strongly interacting ultracold Fermi gases. This enables us to directly observe the smooth transformation from a bosonic to a fermionic superfluid that takes place in the BEC-BCS crossover. We also use our spectra to determine the evolution of the superfluid gap and find excellent agreement with previous experiments and self-consistent $T$-matrix calculations both in the BEC and crossover regime. However, toward the BCS regime a calculation that includes the effects of particle-hole correlations shows better agreement with our data.

Figure 1

Measuring the excitation spectrum of an ultracold Fermi gas using Bragg spectroscopy. (a) Absorption image of a homogeneous Fermi gas trapped in an approximately cylindrical box potential. (b) Sketch of the experimental setup. Two far-detuned laser beams with frequency and wave vector $({\omega }_1,{\overrightarrow{k}}_1)$ and $({\omega }_2,{\overrightarrow{k}}_2)$ are used to create excitations with energy and momentum transfer $\hslash \omega =\hslash {\omega }_1-\hslash {\omega }_2$ and $\hslash q=|\hslash {\overrightarrow{k}}_1-\hslash {\overrightarrow{k}}_2|$ through a two-photon process. (c) Measurement of the dynamic structure factor $S(q,\omega )$ of a unitary Fermi gas. At low energy and momentum transfer, the Goldstone mode of the superfluid manifests itself as a linear phononic mode with a slope that corresponds to the speed of sound $v_s$. Pair breaking excitations occur as a broad continuum, with a clear onset at an energy corresponding to twice the pairing gap $\mathrm{\Delta }$ of the system. For comparison, the expected value of $2\mathrm{\Delta }$ on unitarity [33] is shown as a red dashed line, a numerical calculation of the center of the Goldstone mode is shown as a red solid line [34]. All data shown in this Letter are obtained by averaging over 7–40 individual measurements.

Figure 2

Evolution of the excitation spectrum in the BEC-BCS crossover. (a) In the deep BEC regime, the excitation spectrum follows the Bogoliubov dispersion of an interacting Bose gas, with a linear sound mode at low momenta and a quadratic dispersion of single-molecule excitations at high momenta. (b),(c) When moving into the crossover regime, the compressibility of the system decreases, and consequently the linear branch has a steeper slope and persists to higher momenta. At the same time, the high-momentum part of the dispersion shows a strongly reduced curvature and starts to broaden, which indicates the transition to pair breaking excitations. (d) At the unitary point, there is already a strong pair breaking continuum, which becomes even more pronounced when going further into the BCS regime (e),(f).

Figure 3

Measurements of the collective mode on the BEC (a) and BCS (b) side of the resonance. The black dots show the fitted center of the collective mode for each momentum slice; the black line is a fit according to the equation $\omega =v_{s}q(1+\zeta q^2)$. (c) Speed of sound $v_s$ across the BEC-BCS crossover (blue dots) extracted from the fit to the collective mode. We find good agreement with a previous measurement of the speed of sound performed via fixed-momentum Bragg spectroscopy [26] (light blue stars), a measurement of the Bertsch parameter at unitarity [21] (orange diamond), and a quantum Monte Carlo calculation of the equation of state [58] (dashed line). (d) Prefactor $\zeta$ of the $q^3$ correction to the collective mode. In the BEC regime, the dispersion bends upward and $\zeta >0$. When moving toward the crossover regime, the value of $\zeta$ decreases until it changes sign at an interaction strength of $1/k_{F}a\approx 0.2$. For interaction parameters $1/k_{F}a\lesssim 0.2$, the collective mode bends down and $\zeta <0$. The statistical uncertainties of the data points shown in (c) and (d) are smaller than the marker size.

Figure 4

Measurement of the pairing gap in the BEC-BCS crossover. (a) Heating rate $S(q,\omega )\omega$ on the BCS side of the resonance ($1/k_{F}a=-0.43$) measured at a fixed-momentum transfer of $\hslash q=1.5\hslash k_F$. The onset of the pair breaking continuum is clearly visible in the data; the red line shows a fit of a line shape from quasiparticle random-phase approximation calculations used to extract the value of the pairing gap $\mathrm{\Delta }$ [blue squares in panel (c)] [34]. (b) Close to resonance, we perform measurements at low momentum to separate the onset of the pair breaking continuum from the collective mode. The resulting onset determined from a bilinear fit (red) is shown in panel (c) as blue dots [34]. (c) Pairing gap $\mathrm{\Delta }$ across the BEC-BCS crossover. The error bars denote the $1\sigma$ confidence interval of the fit and are (mostly) smaller than the symbol size. Our data is in good agreement with previous measurements (orange diamonds [24], blue stars [26]). When comparing to theory, we find excellent agreement with self-consistent $T$-matrix calculations [33] close to resonance and in the BEC regime (black solid line), but toward the BCS regime calculations including Gor’kov-Melik-Barkhudarov corrections [59] (light blue line) are closer to our data.