Equation of State of Ultracold Fermions in the 2D BEC-BCS Crossover Region

We report the experimental measurement of the equation of state of a two-dimensional Fermi gas with attractive $s$-wave interactions throughout the crossover from a weakly coupled Fermi gas to a Bose gas of tightly bound dimers as the interaction strength is varied. We demonstrate that interactions lead to a renormalization of the density of the Fermi gas by several orders of magnitude. We compare our data near the ground state and at finite temperature with predictions for both fermions and bosons from quantum Monte Carlo simulations and Luttinger-Ward theory. Our results serve as input for investigations of close-to-equilibrium dynamics and transport in the two-dimensional system.

Figure 1

Low-temperature EOS across the 2D BEC-BCS crossover. The experimental results are obtained from measurements of the quasi-2D gas at the lowest attainable temperatures, which corresponds to $T/T_F\approx 0.05$ and 0.1 on the Bose and Fermi sides. The data points shown as diamonds (circles) correspond to measurements in the superfluid (normal) phase. The solid red line on the Bose side corresponds to the mean field formula $\tilde{\mu }/{ϵ}_F=-{\eta }^{-1}/4$ with $\eta =\ln (k_F{a}_{2\mathrm{D}})$, whereas the dashed and solid green lines on the Fermi side display the non-self-consistent and self-consistent Hartree-Fock predictions $1/(1+{\eta }^{-1})$ and $1-{\eta }^{-1}$ for weakly attractive fermions. The orange line is the prediction for the ground state EOS from recent QMC calculations [48].

Figure 2

Phase space density on the Bose side of the crossover. The experimental data points are shown as filled shapes. We compare to bosonic theory with effective coupling strengths $\tilde{g}=0.60$, 1.07, 2.75. Open shapes represent the EOS extracted from the QMC simulation of the quasi-2D Bose gas trapped in an external potential with similar parameters as employed in experiment. The dashed curves show the classical MC prediction for the weakly coupled homogeneous 2D Bose gas from Ref. [58] extrapolated to large values of $\tilde{g}$. For moderate densities we find good agreement between all three approaches. For large densities, however, experiment and trapped QMC calculations deviate from the classical homogeneous result which scales like the mean field prediction $n_d{\lambda }_d^2=(2\pi /\tilde{g})\beta {\mu }_d+\ln [(2\tilde{g}/\pi )n_d{\lambda }_d^2-2\beta {\mu }_d]$. This may be due to quantum effects appearing at large $\tilde{g}$, as well as the axial confinement.

Figure 3

EOS in the crossover regime shown as the density normalized by the ideal Fermi gas $n_0(\mu ,T)=2{\lambda }_T^{-2}\ln (1+e^{\beta \mu })$. The experimental data points (filled shapes) are compared with the second order virial expansion at low values of $\beta \mu$ (coloured dashed lines). The displayed errors are purely statistical, with systematic uncertainties estimated at 13%–15% and given for each data set in Table IV in the Supplemental Material [36]. We compare our results with theoretical predictions in the 2D BEC-BCS crossover from Luttinger-Ward theory ([23], solid black lines) and fermionic QMC simulations ([24], dotted black lines), with the corresponding values of $\beta {ϵ}_B$ labeling each curve. Note that the vertical scale differs by a factor of 10 in each panel.