Pair fraction in a finite-temperature Fermi gas on the BEC side of the BCS-BEC crossover

We investigate pairing in a strongly interacting two-component Fermi gas with positive scattering length. In this regime, pairing occurs at temperatures above the superfluid critical temperature; unbound fermions and pairs coexist in thermal equilibrium. Measuring the total number of these fermion pairs in the gas we systematically investigate the phases in the sectors of pseudogap and preformed pair. Our measurements quantitatively test predictions from two theoretical models. Interestingly, we find that already a model based on classical atom-molecule equilibrium describes our data quite well.

Figure 1

Theoretical phase diagram for a balanced two-component harmonically trapped ultracold Fermi gas in the vicinity of a Feshbach resonance (vertical line) where $k_F$ and $T_F$ are determined in the trap center. Shown are calculated contours for various pair fractions. Dotted lines are based on a self-consistent $t$-matrix approach [23], while solid lines are based on a classical model of noninteracting atoms and molecules (see text) [22]. Close to the Feshbach resonance the solid lines are blurred because the classical model is expected to lose its validity. The cyan dash-dotted line marks a pair breaking temperature, as calculated by [24] with a BCS mean-field model that was extended to the near-BEC regime. The gray shaded area marks the superfluid phase below the critical temperature $T_c$ which was calculated within the self-consistent $t$-matrix approach [25].

Figure 2

Measurement of the number of fermion pairs. (a) and (b) Optical transfer method. A resonant laser pulse transfers pairs to states which are invisible to our detection scheme [blue arrows (1)]. The total number $N_{\sigma }\left(\mathrm{\Delta }t\right)$ of remaining fermion pairs and single atoms is measured by absorption imaging [red arrows (2)]. (b) $N_{\sigma }\left(\mathrm{\Delta }t\right)/N_{\sigma }\left(0\right)$ as a function of the pulse width $\mathrm{\Delta }t$ at a magnetic field of $726\mathrm{G}$ for various temperatures $T/T_{\mathrm{F}}=\{0.64,0.79,1.2,1.4,1.7\}$. The solid lines are fit curves using Eq. (1). (c) and (d) Magnetic transfer method. Using absorption imaging, the particle number $N_{\sigma }=N_a+N_p$ is measured at the magnetic field (1) and the number of unbound atoms $N_a$ is measured after a fast ramp to (2). (d) The measured particle numbers at (1) ($B=726\mathrm{G}$, green solid circles) and at (2) ($B=550\mathrm{G}$, red solid squares) for various temperatures $T/T_{\mathrm{F}}$.

Figure 3

Measured pair fractions $N_p/N_{\sigma }$ (blue circles) at $726\mathrm{G}$ for various temperatures $T/T_{\mathrm{F}}$. (a) Optical transfer (OT) method; (b) magnetic transfer (MT) method (see Fig. 2). We note that due to evaporative cooling ${\left(k_{\mathrm{F}}a\right)}^{-1}$ also changes with $T/T_{\mathrm{F}}$ (orange diamonds). The green curves are calculations based on the classical model.

Figure 4

Map of the pair fraction $N_p/N_{\sigma }$ as a function of temperature and interaction strength on the BEC side of the Feshbach resonance. The circles (diamonds) are measurements obtained with the OT (MT) method. The thick solid and dashed lines are classical model calculations (cf. Fig. 1). They are dashed in the strong-interaction regime where the classical model is expected to be no longer valid. The error bars include both a statistical and a systematic part, i.e., the standard deviation of the mean of 10 temperature measurements and the uncertainty in determining the molecule fraction from the fit, respectively. The upper-right area bounded by the gray dash-dotted line exhibits $>5\%$ particle loss due to inelastic collisions on the time scale of a measurement. The gray shaded area indicates the superfluid phase below $T_c$, as in Fig. 1.

Figure 5

Ratio $N_{\sigma }\left(\mathrm{\Delta }t\right)/N_{\sigma }\left(0\right)$ after an optical transfer pulse of length $\mathrm{\Delta }t$ at a magnetic field of $820\mathrm{G}$ for various temperatures (see legend). The solid lines are fits of an exponential decay towards a constant offset.