Spin-polarized fermions with $p$-wave interactions

We have created quantum degenerate Fermi gases of ${}^6\mathrm{Li}$ atoms at unprecedented high densities exceeding $1/{\lambda -}^3$ (where $\lambda$ is the wavelength of resonant light). In this regime, new optical properties are predicted. Spin-polarized Fermi gases are usually regarded as noninteracting due to the weakness of $p$-wave interactions. Here we study the properties of “$p$-wave matter,” where elastic and inelastic $p$-wave collisions determine the dynamics of the system. We characterize thermalization, evaporative cooling, and inelastic two-body and three-body collisions. $P$-wave dipolar relaxation creates a metastable mixture of the lowest and highest hyperfine states.

Figure 1

The ratio of good (elastic) to bad (inelastic) collisions, and the elastic collision rate for ${}^6\mathrm{Li}$ atoms near zero field, far away from the $p$-wave Feshbach resonance. The solid line shows the ratio for an harmonically trapped cloud with $T=T_F$ assuming a vacuum lifetime of 60 seconds. The dashed line is the elastic collision rate. In this work, we have explored the shaded region.

Figure 2

Three-body loss for spin-polarized fermions. Different temperatures are realized by reducing the trap laser power. The three-body loss coefficient $L_3$ scales quadratically with temperature. The black line is a parabolic fit through the origin, resulting in $L_3=(3.55\pm 0.22)\times {10}^{-23}\times T^2$. Error bars show the statistical one standard deviation uncertainties of the fit parameter $L_3$ used to describe the atom number decay. The inset shows the decay curve at $T=310\mu \mathrm{K}$ and its fit to Eq. (1).

Figure 3

Observation of cross-dimensional thermalization. Normalized temperature difference $\mathrm{\Delta }T/T_{\text{ave}}$ between the radial and axial directions as a function of hold time after creating a sample with anisotropic energy distribution. The dashed line is an exponential fit $\mathrm{\Delta }T/T_{\text{ave}}=Aexp(-{\mathrm{\Gamma }}_{\mathrm{th}}t)+c$. The temperature difference has a small (5%) offset in all measurements the reason for which we have not tracked down (possibly due to anisotropic heating/cooling or nonsudden switch-off of potentials to initiate ballistic expansion). Inset shows the average temperature, demonstrating that there is no heating during the measurement. Error bars represent the standard deviation of the data averaged over three measurements.

Figure 4

Determination of the background $p$-wave scattering length. Thermalization rates ${\mathrm{\Gamma }}_{\text{th}}$ were measured for a range of atom numbers and temperatures and are shown as a function of the mean density times $T^{5/2}$, which is the scaling with density and temperature given by Eq. (2). A linear fit weighing the data with inverse of variances (vertical error bars) gives the $p$-wave background scattering length $|V_p|={\left({39}_{-1.6}^{+1.3}{a}_0\right)}^3$. The dashed lines represent the 95% confidence interval of the fit. We note that the only fit parameter is the slope, representing $V_p$.

Figure 5

Evaporative cooling using $p$-wave collisions. (a) Time-of-flight absorption image (average of 4 shots) of degenerate atoms at $T/T_F=0.42$. Temperature is determined from a Gaussian fit to the wings of the cloud, and Fermi energy is calculated using measured number of atoms and trapping frequencies [33]. (b) Evolution of $T/T_F$ for different final trap depths and therefore different final numbers of atoms. [(c) and (d)] Comparison of the azimuthally averaged radial cloud profiles, before (c) and after (d) introducing a second spin component of state $|1/2,-1/2\rangle \equiv |2\rangle$ which undergoes $s$-wave collisions with state $|1\rangle$. Red lines show the Fermi-Dirac profile set by the atom number, temperature and trapping frequencies. The cloud in (c) is not in thermal equilibrium, in contrast to (d). Note that the number of atoms in (d) is lower, probably due to enhanced three-body loss.

Figure 6

Dipolar relaxation of state $|6\rangle$, leading to a population of state $|1\rangle$. Population of both states, as a function of hold time in the trap at full power. Lines are a guide to the eye. The initial number of state $|1\rangle$ atoms are generated while the optical trap was ramped up to full power.

Figure 7

The ratio of good to bad collisions (a) and the elastic collision rate (b) near the $p$-wave Feshbach resonance at 159 G for a harmonically trapped cloud with $T=T_F$.

Figure 8

Comparison between $s$- and $p$-wave collision rates. (a) shows the rate of collisions involving an atom with speed $v$ with respect to the thermally averaged collision rate. In (b), the collision rates are weighted by the Maxwell-Boltzman speed distribution. (c) shows the ratio of $s$- and $p$-wave collisions for an atom at speed $v$. (d) is the fraction of collisions up to a kinetic energy of $\eta k_{B}T$, where $\eta$ is the truncation parameter.