Condensation energy of a spin-1/2 strongly interacting Fermi gas

We report a measurement of the condensation energy of a two-component Fermi gas with tunable interactions. From the equation of state of the gas, we infer the properties of the normal phase in the zero-temperature limit. By comparing the pressure of the normal phase at $T=0$ to that of the low-temperature superfluid phase, we deduce the condensation energy, i.e., the energy gain of the system upon being in the superfluid rather than the normal state. We compare our measurements to a ladder approximation description of the normal phase and to a fixed-node Monte Carlo approach, finding excellent agreement. We discuss the relationship between condensation energy and pairing gap in the BEC-BCS crossover.

Figure 1

Reduced pressure $h(t,b)=P({\mu }_1,{\mu }_2,T)/2P_0\left(\overline{\mu }\right)$ of the spin-1/2 unitary Fermi gas, where $P_0$ is the $T=0$ Fermi pressure of an ideal gas, $t$ the reduced temperature $k_{B}T/\mu$, and $b=0$ the unpolarized gas. Open black circles are data from [13] taken at $B=834$ G, while filled black circles include a small correction due to a recently determined downshift of the Feshbach resonance [10]. This correction is estimated using Tan's contact calculated by the bold diagrammatic Monte Carlo (bDMC [16]) (see Appendix B for details). The Fermi-liquid fit is shown as the solid red line, and the extrapolated zero-temperature pressure of the normal state ${\xi }_N^{-3/2}$ is represented by the (red) $\mathsf{X}$. MIT data from [15] are represented by (blue) squares; the corresponding fit, by the dashed (blue) line; and the extrapolation at $t=0$, by the open (blue) square. The bDMC calculation [16] is shown by the solid green line. The agreement with the bDMC data is excellent, while a small discrepancy from the MIT data is visible near the superfluid-to-normal transition around $t_c=0.40$ [15] or $t_c=0.33$ here [17] (the latter represented by the dashed vertical line). The dashed horizontal (red) line corresponds to the superfluid pressure; the dotted black line, to the ideal gas.

Figure 2

Pressure of the spin-imbalanced gas in the BEC-BCS crossover at $t=0$. The position of the first-order phase transition to the superfluid is shown by the vertical dashed black line. (a) Unitary limit. The Fermi-liquid fit is shown by the solid red line; the $t=0$ equation of state in the superfluid phase, by the solid horizontal blue line. The pressure of the noninteracting gas is displayed as the dotted gray line. The $t=0$ and $b=0$ extrapolation of the normal phase pressure is shown by the (red) $\mathsf{X}$; the condensation pressure, by the double-arrows. (a–c) Results of the ladder approximation for the normal phase are shown in green for $\delta =0$, $-0.58$, and $+0.2$, respectively.

Figure 3

Pressure of the normal $h_N$ [open (red) squares] and superfluid $h_S$ [filled (blue) circles] phases at low temperature in the BEC-BCS crossover measured in [14]. The thick lower solid (green) line is the result of the ladder approximation. The upper solid (blue) line is a guide for the eye, while the lower solid (red) line is the result of fixed-node Monte Carlo calculations [18]. The difference between the blue and the red or green lines is the condensation pressure.

Figure 4

Relation between the condensation energy and the superfluid pairing gap. Dimensionless condensation energy ${\xi }_N-{\xi }_S$ versus interaction strength $1/k_{F}a$ extracted from the $b\to 0$ extrapolation (solid black line). Filled (red) circles represent the BCS expression, (8), using the values of $\Delta$ measured in [28]. The prediction from mean-field BCS theory is shown by the dashed (blue) line, and the open (green) circle with a vertical bar is the $t\to 0$ extrapolation of Fig. 1. A fixed-node Monte Carlo calculation [18] coincides with the solid black line. Inset: Rratio of the condensation energy ${\xi }_N-{\xi }_S$ to $\frac{5}{8}{\left(\frac{\Delta }{E_F}\right)}^2$.

Figure 5

Dimensionless pressure of the spin-imbalanced gas in the unitary limit ($\delta =0$). Experimental results are shown as filled black circles. The theory using Eq. (6) (Landau's method) is shown by the thick solid (green) line, while the result of Eqs. (A1) and (A2) is shown as the dashed (red) line. The horizontal (blue) line shows the value of the dimensionless pressure in the superfluid state.

Figure 6

Pressure of the unpolarized unitary Fermi gas. The original data taken at 834 G are shown as open black circles [13], while the corrected EoS at 832.18 G is displayed as filled black circles (see text). Measurements from MIT and Tokyo are shown as filled (blue) squares [15] and open (red) triangles [31], respectively, and the bDMC calculation from Amherst, as the solid (green) line [16]. At the lowest temperatures, we find a corrected Bertsch parameter, ${\xi }_S=0.40\left(2\right)$.