Phase diagram of a two-component Fermi gas with resonant interactions

I Appendix

The experimental procedure has been described in previous publications Zwierlein et al. (2006a, b); Shin et al. (2006). A degenerate Fermi gas of 6Li atoms was first prepared in an optical trap, using laser cooling and sympathetic cooling with 23Na atoms. A variable spin mixture of the two lowest hyperfine states $|\uparrow\rangle$ and $|\downarrow\rangle$ (corresponding to the $|F=1/2,m_{F}=1/2\rangle$ and $|F=1/2,m_{F}=-1/2\rangle$ states at low magnetic field) was created at a magnetic field $B=885$ G. The final evaporative cooling by lowering the trap depth and all measurements were performed at $B=833$ G. The temperature of the cloud was controlled by the lowest value of the trap depth in the evaporative cooling process. The axial trap frequency was $f_{z}=23$ Hz. The two transverse trap frequencies are equal within less than $2\%$.

The optical signal in the phase-contrast imaging is proportional to the net phase shift of the imaging beam passing through a Fermi mixture, i.e. $c_{\uparrow}n_{\uparrow}-c_{\downarrow}n_{\downarrow}\propto n_{\uparrow}/(\nu-\nu^{0}_{\uparrow})+n_{\downarrow}/(\nu-\nu^{0}_{\downarrow})$, where $\nu$ is the probe frequency of the imaging beam, and $\nu^{0}_{\uparrow}$ and $\nu^{0}_{\downarrow}$ are the resonance frequencies of the optical transition for the states $|\uparrow\rangle$ and $|\downarrow\rangle$, respectively. When the probe beam is tuned to the middle of the two transitions, i.e. $\nu=\nu_{0}=(\nu^{0}_{\uparrow}+\nu^{0}_{\downarrow})/2$, the optical signal reflects the density difference $n_{d}=n_{\uparrow}-n_{\downarrow}$ with $c_{\uparrow}=c_{\downarrow}$. In our experiment, two phase-contrast images of the same sample were taken consecutively with different probe frequencies, $\nu_{1}$ and $\nu_{2}$. The two images record the density difference $n_{d1}=n_{\uparrow}-n_{\downarrow}$ and the weighted density difference $n_{d2}=\alpha_{\uparrow}n_{\uparrow}-\alpha_{\downarrow}n_{\downarrow}$. $\nu_{1}$ was determined by zeroing the optical signal with an equal mixture and $\alpha_{\uparrow,\downarrow}$ was determined by the signal ratio between the first and the second image for a highly imbalanced Fermi mixture with $|\delta|>95\%$ (an almost fully polarized gas). Finally, we obtained $n_{\uparrow}=(\alpha_{\downarrow}n_{d1}-n_{d2})/(\alpha_{\downarrow}-\alpha_{\uparrow})$ and $n_{\downarrow}=(\alpha_{\uparrow}n_{d1}-n_{d2})/(\alpha_{\downarrow}-\alpha_{\uparrow})$. The difference between $\nu_{1}$ and $\nu_{2}$ was chosen to lie between 8 and 13 MHz. The time interval between the two images was $10~{}\mu$s, and the pulse duration of each probe beam was $15~{}\mu$s. Because the probe beam was off-resonant, no heating effect of the first pulse was observed in the second image.

Low-noise profiles were obtained by averaging the column density distribution of phase-contrast images along elliptical equipotential lines ($\lambda^{2}x^{2}+z^{2}=r^{2}$). For the measurement of the critical spin polarization, the averaging region was restricted to $|x|<12~{}\mu$m in order to preserve the sharp features at the phase boundary. The diffraction limit for our imaging system was about $2~{}\mu$m. For the determination of local quantities in the profiles, we averaged over $\pm 5~{}\mu$m around a given position. For temperature determination, the averaging region was restricted to an axial sector of $\pm 60^{\circ}$ to avoid corrections due to radial anharmonicities (see appendix). The relative temperature $T^{\prime}$ is determined as $T^{\prime}\equiv T/T_{F0}=(-6Li_{3}(-\zeta))^{-1/3}$, where $\zeta$ is the fugacity obtained from the fit ($Li_{s}(z)\equiv\sum_{k=1}^{\infty}z^{k}/k^{s}$ is the Poly-Logarithmic function of order $s$).

In our previous work Zwierlein et al. (2006b); Shin et al. (2006), temperatures have been determined by fitting the spatial wings of the majority component after expansion. However, we found that one can neglect collisions with the minority atoms in the core only for large population imbalances. In a simplified picture, one can regard collisions with the inner core as collisions with a moving wall, which moves outward radially and inward axially (due to the magnetic trapping potential). This results in different average kinetic energies (transversely and axially) of the free majority atoms in the outer region. Figure 7 shows the density distribution of the majority and minority components after expansion. Although the temperature has been overestimated by only $20\%$ for typical experimental conditions ($\delta\approx 60\%$) in refs Zwierlein et al. (2006b); Shin et al. (2006), we do not regard this technique as well-calibrated absolute thermometry.

One other concept for thermometry determines temperature as the derivative of entropy with energy. So far, this concept could be implemented only for balanced fermion mixtures with certain approximations, and due to the need of determining a derivative, could only be used to obtain temperatures averaged over a certain range Luo et al. (2007).

For the determination of temperatures from the spatial in situ profiles it was necessary to address the anharmonicity of the trapping potential. Our trap is generated by a weakly focused (beam waist $w\approx 125~{}\mu$m) infrared Gaussian laser beam (wavelength 1064 nm) near the saddle point of a magnetic potential. The total trapping potential is given as

where $\rho^{2}=x^{2}+y^{2}$. Note that gravity has been compensated by a magnetic field gradient. The axial confinement comes mainly from the magnetic potential with oscillation frequency of $f_{z}=23$ Hz. The transverse magnetic potential is anti-trapping and limits the trap depth as

where $f_{\rho}$ is the transverse oscillation frequency in the central harmonic region. When the trap depth is comparable to the Fermi energy of a sample, the transverse anharmonicity will affect the shape of the cloud. Although in our experiments, the inner core and the outer cloud had the same aspect ratio as the trapping potential, indicating the absence of anharmonic effects, anharmonicities were not negligible in the spatial wings used to determine the temperature.

This issue was addressed by adjusting the angular averaging region (Fig. 8). Since the trapping potential is only anharmonic for large $\rho$, we could reduce the effect by decreasing the angle of the averaging sector around the $z$ direction. Both the experimental data and an exact simulation for an ideal Fermi gas show that the fitted temperature remains almost constant up to a certain angle and then increases when the averaging sector includes more of the transversely outer region. In our temperature determination, we chose the averaging sector to be $\pm 60^{\circ}$ which was large enough to create low-noise profiles, but kept the effect of the anharmonicities to below $10\%$. The 1D fit to angularly averaged profiles was computationally more efficient than a 2D fit to a selected region of the image. In a 2D fit, one could also include anharmonic terms in the fitting function.

Polarized superfluid at finite temperature

When the two spin components have a chemical potential difference $2h$, the BCS-type superfluid has two branches of quasiparticles with excitation energies $\sqrt{(\epsilon_{k}-\mu)^{2}+\Delta^{2}}\pm h$ where $\epsilon_{k}=\hbar^{2}k^{2}/2m$. At finite temperature, the superfluid is polarized due to the large thermal population of the lower branch compared to the upper branch. An interesting situation arises when $h$ becomes larger than $\Delta$, i.e. the lower branch has negative energy quasiparticles, implying that even at zero temperature the superfluid state would have a finite polarization. Our experiments show $h_{c}<\Delta$ at very low temperature, suggesting that a polarized superfluid state exists only at finite temperature. The breached-pair state with $h_{c}>\Delta$ at zero temperature has been predicted in a stronger coupling region (on the BEC side of the Feshbach resonance). Since $\Delta$ gradually decreases with higher temperature, it might be possible to have $h_{c}>\Delta$ at finite temperature, at least in the weakly-interacting BCS limit where $\Delta$ smoothly approaches zero at a second order phase transition point. One interesting problem is identifying this gapless region of $h>\Delta$ in the phase diagram for various coupling regimes.

References