Spin-energy correlation in degenerate weakly interacting Fermi gases

Weakly interacting Fermi gases exhibit rich collective dynamics in spin-dependent potentials, arising from correlations between spin degrees of freedom and conserved single-atom energies, offering broad prospects for simulating many-body quantum systems by engineering energy-space “lattices,” with controlled energy landscapes and site-to-site interactions. Using quantum-degenerate clouds of ${}^6\mathrm{Li}$, confined in a spin-dependent harmonic potential, we measure complex, time-dependent spin-density profiles, varying on length scales much smaller than the cloud size. We show that a one-dimensional mean-field model, without additional simplifying approximations, quantitatively predicts the observed fine structure. We measure the magnetic fields where the scattering lengths vanish for three different hyperfine state mixtures to provide constraints on the collisional (Feshbach) resonance parameters.

Figure 1

Spin-energy correlation produces spin segregation in a degenerate Fermi gas with an $s$-wave scattering length of 5.2 bohrs. The palettes are $50\times 950 \mu {\mathrm{m}}^2$. Left to right: $n_1, n_2, n_1-n_2$, and $n_1+n_2$ in units of ${(n_1+n_2)}_{\max }$ at $t=0$ (upper) and $t=800$ ms (lower) after coherent excitation of a $|1\rangle -|2\rangle$ superposition state. Note that $n_1-n_2$ evolves in time while $n_1+n_2$ remains constant, due to single-particle energy conservation.

Figure 2

Spin-density profiles for a degenerate ($T/T_F=0.35$) Fermi gas at $t=800\mathrm{ms}$ relative to coherent excitation. Data (blue dots) versus prediction (red curves) showing quantitative agreement. Left to right: $n_1, n_2, n_1-n_2$, and $n_1+n_2$ in units of the peak total density. Each solid curve is the mean-field model with a fixed scattering length of $a=3.04$ bohrs ($B=528.147\mathrm{G}$) and a fitted cloud size ${\sigma }_{Fx}\equiv \sigma =329\mu \mathrm{m}$, obtained by fitting the total density $n_1+n_2$ to a one-dimensional (1D) Thomas-Fermi profile [Eq. (B38)].

Figure 3

Spin-density profiles in a degenerate sample $T/T_F=0.35$ at selected times relative to coherent excitation. $\mathrm{\Delta }n\left(0\right)=n_1(0,t)-n_2(0,t)$ is given in units of $n_1\left(0\right)+n_2\left(0\right)$. Solid curves: Mean-field model with the same scattering length for each time and a fitted cloud size within a few percent of the measured average value, $\sigma =322.0\left(1.5\right)\mu \mathrm{m}$. Top three panels: $B=528.817\mathrm{G}, a=5.17a_0$. Bottom three panels: $B=525.478\mathrm{G}, a=-5.39a_0$. Note that the spin density inverts when the scattering length changes sign.

Figure 4

Central spin density versus evolution time for various magnetic fields near the zero crossing of the $|1\rangle -|2\rangle$ scattering length. $\mathrm{\Delta }n\left(0\right)=n_1(0,t)-n_2(0,t)$ is given in units of $n_1\left(0\right)+n_2\left(0\right)$. Solid curves show the mean-field model with the scattering length $a$ as a fit parameter. The fitted values of $a$ are plotted in Fig. 7.

Figure 5

Decay of the amplitude of the central spin density versus time for $a=-14.9a_0$. The dashed curve shows the predicted amplitude for the average density. The red curve shows the average of the predictions based on the measured atom numbers and cloud widths for each shot.

Figure 6

Spin-density profiles versus time for $a=-14.9a_0$ versus predictions (red curves) with the same scattering length for each time and a fitted cloud size within a few percent of the measured average value, $\sigma =330.6\mu \mathrm{m}$.

Figure 7

Fitted scattering length $a$ versus measured magnetic field for a $|1\rangle -|2\rangle$ mixture ($a_0=1\mathrm{bohr}$). Error bars denote one standard deviation, obtained for each ${\chi }^2$ fit of Fig. 4.

Figure 8

Measurement of the zero crossing field for a degenerate ${}^6\mathrm{Li} |1\rangle -|2\rangle$ mixture. The plots show the change in cloud size between $t=0$ and $t=800$ ms for state 1 (squares), state 2 (diamonds), and the difference in the cloud sizes of the two spin states at $t=800$ ms (circles). Solid lines are corresponding linear fits, crossing zero (dashed line) when $a=0$. Error bars denote the standard deviation of the mean of five runs.

Figure 9

High-temperature spin-density profile of a $|1\rangle -|2\rangle$ mixture for $t=400$ ms. $T=45.7\mu \mathrm{K}$ and $B=527.466\mathrm{G}$, where the zero-energy $s$-wave scattering length is $0.90\mathrm{bohr}$. Here ${\sigma }_G=323\mu \mathrm{m}$ is the Gaussian $1/e$ radius of the total density profile.

Figure 10

Tuning rate of the scattering length $a$ of a $|2\rangle -|3\rangle$ mixture versus measured magnetic field ($a_0=1\mathrm{bohr}$) Error bars denote one standard deviation, obtained for each ${\chi }^2$ fit to the time-dependent central amplitude for the given $B$.

Figure 11

Measurement of the zero crossing field for a degenerate ${}^6\mathrm{Li} |2\rangle -|3\rangle$ mixture. The plots show the change in cloud size between $t=0$ and $t=800$ ms for state 3 (squares), state 2 (diamonds), and the difference in the cloud sizes of the two spin states at $t=800$ ms (circles). Solid lines are corresponding linear fits, crossing zero (dashed line) when $a=0$. Error bars denote the standard deviation of the mean of five runs.

Figure 12

Measurement of the zero crossing field for a degenerate ${}^6\mathrm{Li} |1\rangle -|3\rangle$ mixture. The plots show the change in cloud size between $t=0$ and $t=800$ ms for state 3 (squares) and state 1 (diamonds). Solid lines are corresponding linear fits, crossing zero (dashed line) when $a=0$. Error bars denote the standard deviation of the mean of five runs.

Figure 13

Spin-density profiles (blue dots) for a degenerate sample $T/T_F=0.35$ versus evolution time relative to coherent excitation. Each data profile is the average of five runs, taken in random time order. Each solid red curve is the mean-field model with a fixed scattering length of $a=5.23$ bohrs ($B=528.844\mathrm{G}$) and a fitted cloud size within a few percent of the average value $\sigma =329\mu \mathrm{m}$.