Quantitative comparison between theoretical predictions and experimental results for Bragg spectroscopy of a strongly interacting Fermi superfluid

Theoretical predictions for the dynamic structure factor of a harmonically trapped Fermi superfluid near the Bose-Einstein condensate–Bardeen-Cooper-Schrieffer (BEC-BCS) crossover are compared with recent Bragg spectroscopy measurements at large transferred momenta. The calculations are based on a random-phase (or time-dependent Hartree-Fock-Gorkov) approximation generalized to the strongly interacting regime. Excellent agreement with experimental spectra at low temperatures is obtained, with no free parameters. Theoretical predictions for zero-temperature static structure factor are also found to agree well with the experimental results and independent theoretical calculations based on the exact Tan relations. The temperature dependence of the structure factors at unitarity is predicted.

Figure 1

Quantitative comparison of theoretical and experimental Bragg spectra [see Eq. (6)]. The RPA prediction (lines) agrees well with the experimental data (empty squares) [5] at the BEC-BCS crossover, with no free parameters. The spectrum is normalized so that the area below the curve is unity. The frequency is measured in units of the recoil energy of the atoms (see text).

Figure 2

Zero temperature spin parallel $S_{\uparrow \uparrow }(\mathbf{q},\omega )$ (dashed lines), antiparallel $S_{\uparrow \downarrow }(\mathbf{q},\omega )$ (dot-dashed lines), and total dynamic structure factor $S(\mathbf{q},\omega )=2[S_{\uparrow \uparrow }+S_{\uparrow \downarrow }]$ (solid lines) across the BEC-BCS crossover: $1/k_{F}a=0.5$ (a), $0.0$ (b), and $-0.5$ (c). The negative weight in $S_{\uparrow \downarrow }$ at about the recoil energy is consistent with the exact sum rule $\int \omega S_{\uparrow \downarrow }(\mathbf{q},\omega )d\omega =0$ [28].

Figure 3

Quantitative comparison between theory and experiment for the zero temperature static structure factor across the crossover. For $S{}_{\uparrow \downarrow }$, with no free parameters our RPA prediction (plus symbols) agrees well with the experimental data for $S(\mathbf{q},\omega )-1$ (solid circles with error bars) [35] and an independent theoretical result based on the exact Tan relations (solid line) [36]. At large transferred momentum, $S{}_{\uparrow \uparrow }\simeq 1$. The inset highlights the RPA prediction with respect to the Tan-relation result in the BCS regime.

Figure 4

Temperature dependence of the dynamic (a) and static (b) structure factor for a unitary Fermi gas in harmonic traps at $q=5k_F$. According to the Tan relation, $S_{\uparrow \downarrow }(\mathbf{q})\simeq 128\zeta /[175{\xi }^{1/4}(q/k_F)]$ at $T=0$ [35], where $\xi$ and $\zeta$ are the universal parameters at unitarity [17]. The symbols in (b) show predictions using theoretically or experimentally determined $\xi$ and $\zeta$ : ENS experiment (square) [17], Gaussian pair fluctuation theory (circle) [12], and self-consistent theory (triangle) [37]. Here, $T_c\simeq 0.37T_F=0.37E_F/k_B$.