Pairing in Few-Fermion Systems with Attractive Interactions

We study quasi-one-dimensional few-particle systems consisting of one to six ultracold fermionic atoms in two different spin states with attractive interactions. We probe the system by deforming the trapping potential and by observing the tunneling of particles out of the trap. For even particle numbers, we observe a tunneling behavior that deviates from uncorrelated single-particle tunneling indicating the existence of pair correlations in the system. From the tunneling time scales, we infer the differences in interaction energies of systems with different number of particles, which show a strong odd-even effect, similar to the one observed for neutron separation experiments in nuclei.

Figure 1

(a) To study a system of two attractively interacting fermions, we study its tunneling dynamics by creating a barrier of fixed height and recording the number of particles remaining in the trap after different hold times. We find that the tunneling slows down as we increase the strength of the interparticle interaction from zero (blue squares) to intermediate (orange dots) or to larger values (red triangles), where the solid lines are fits according to the tunneling model described in the text and the errors are the standard error of the mean of 100 individual measurements. This shows the increase of the effective barrier height due to the energy shift caused by the attractive interaction. (b) Coupling constant $g_{\uparrow \downarrow }$ in the untilted potential as a function of the magnetic field with $a_{\parallel }=\sqrt{\hslash /\mu {\omega }_{\parallel }}$. Note that $g_{\uparrow \downarrow }$ depends on the parameters of the deformed potential and is therefore given in units of $a_{\mathrm{ref}}\hslash {\omega }_{\mathrm{ref}}$ for the different measurements [16].

Figure 2

Measured time evolution of the probabilities $P_1(t)$ and $P_2(t)$ to find one (green dots) or two (blue triangles) particles in the tilted potential at an intermediate interaction strength of $g_{\uparrow \downarrow }=-0.64$. The blue solid line shows a fit of Eq. (5) to $P_2(t)$ with free parameters $f$ and ${\gamma }_2$. The green line shows the fit to $P_1(t)$ with the single free parameter ${\gamma }_p$, where the shaded region indicates the uncertainty that results from our determination of the shape of the trapping potential. For comparison, the dashed line shows the result from Eq. (6) with ${\gamma }_p$ set to 0, which is also consistent with our data. The errors are the 68% confidence interval of about 100 individual measurements. The inset shows a sketch of the loss processes included in our tunneling model: subsequent single-particle tunneling with rates ${\gamma }_s$ and ${\gamma }_{s_0}$ and direct pair tunneling with a rate ${\gamma }_p$.

Figure 3

Probability of finding a single particle in the trap plotted as a function of the mean particle number $\overline{N}$. The solid lines show the results of fits to the data according to our tunneling model (see Fig. 2). For a noninteracting system, the probability follows the expectation value of completely uncorrelated tunneling given by the black dashed line. For increasing interaction strengths, it becomes less likely to find a single particle in the trap, which we interpret as a result of increased pair correlations. To quantify this effect, we plot the interaction energy up to intermediate interaction strength as a function of $g_{\uparrow \downarrow }$ in the right panel. Here, pair tunneling is not considered in the analysis as it does not seem to play an important role at these interaction strengths.

Figure 4

Separation energies for systems of $N=1,\dots ,6$ particles at an interaction strength of $g_{\uparrow \downarrow }\approx -0.6$. The small difference in $g_{\uparrow \downarrow }$ for the different particle numbers is due to the dependence of the coupling strength on the depth of the optical potential. The separation energies are normalized by the level spacing of the two uppermost trap states. The error bars show the relative uncertainty originating from the statistical errors of the fits of ${\gamma }_N$ and ${\gamma }_{s_0|\uparrow ⟩}$. The arrows indicate the contributions $E_{ii}$ and $E_{ij}$ of the intrashell (blue) and intershell (green) interaction to the separation energy.