Observation of the Leggett-Rice Effect in a Unitary Fermi Gas

We observe that the diffusive spin current in a strongly interacting degenerate Fermi gas of ${}^{40}\mathrm{K}$ precesses about the local magnetization. As predicted by Leggett and Rice, precession is observed both in the Ramsey phase of a spin-echo sequence, and in the nonlinearity of the magnetization decay. At unitarity, we measure a Leggett-Rice parameter $\gamma =1.08(9)$ and a bare transverse spin diffusivity $D_0^{\perp }=2.3(4)\hslash /m$ for a normal-state gas initialized with full polarization and at one-fifth of the Fermi temperature, where $m$ is the atomic mass. One might expect $\gamma =0$ at unitarity, where two-body scattering is purely dissipative. We observe $\gamma \to 0$ as temperature is increased towards the Fermi temperature, consistent with calculations that show the degenerate Fermi sea restores a nonzero $\gamma$. Tuning the scattering length $a$, we find that a sign change in $\gamma$ occurs in the range $0<(k_{F}a{)}^{-1}\lesssim 1.3$, where $k_F$ is the Fermi momentum. We discuss how $\gamma$ reveals the effective interaction strength of the gas, such that the sign change in $\gamma$ indicates a switching of branch between a repulsive and an attractive Fermi gas.

Figure 1

The Leggett-Rice effect. (a) In a transverse spin spiral along $x_j$, the gradient ${\nabla }_j\mathit{M}\perp \mathit{M}$ drives a spin current ${\mathit{J}}_j\perp \mathit{M}$, as described by Eq. (1). For $\gamma \ne 0$, ${\mathit{J}}_j$ is rotated around $M$ by $\arctan (\gamma )$ compared to $(J_j{)}_{\gamma =0}$. In a spin-echo experiment, this causes both a slower decay of amplitude, $A=|M_x+i{M}_y|$ shown in (b), as well as an accumulated phase, $\varphi =-\arg (i{M}_x-M_y)$ shown in (c). The case of $\theta =5\pi /6$ and full initial polarization is plotted. Dashed lines in (b) and (c) show $\gamma =0$, and gray lines show steps of 0.2 up to $\gamma =\pm 1$.

Figure 2

(a) Amplitude $A$ and phase $\varphi$ (inset) of the $xy$ magnetization measured at unitarity for $(T/T_F{)}_i\simeq 0.2$ and with $M_z=0.00(5)$ (black circles), $M_z=0.74(2)$ (blue), and $M_z=-0.54(3)$ (red). All data are taken at a spin-echo time. Error bars represent uncertainties from the fit to a full interferometric fringe. (b) Plot of $\varphi (t_e)$ versus $M_z\ln [A(t_e)/A_0]$ for the two cases where $M_z\ne 0$. The solid line is a linear fit to both data sets that is used to extract $\gamma$. Error bars represent combined uncertainties from fit and extrapolation. The solid lines in (a) represent fits with Eq. (3) using the value of $\gamma$ obtained by the analysis presented in (b).

Figure 3

Spin transport at unitarity. (a) The measured LR parameter $\gamma$, (b) diffusivity $D_0^{\perp }$, and (c) the ratio ${\lambda }_0=-\hslash \gamma /(2m{D}_0^{\perp })$ are shown versus the initial reduced temperature $(T/T_F{)}_i$. Solid points are each from a phase-sensitive measurement as shown in Fig. 2. Horizontal and vertical error bars represent statistical and fit uncertainties. For these data, $N$ ranges from $50(5)\times 10^3$ at low temperature to $18(4)\times 10^3$ at high temperature. Open circles are results from a fit of Eq. (3) to $\theta \simeq \pi /2$ data such as the black circles in Fig. 2, and also to data from Ref. [3]. Here, we fix $M_z$ and vary $\gamma$ (chosen a posteriori to be non-negative) and $D_0^{\perp }$. Although the two methods provide similar values on average, the phase-sensitive measurements provide reduced scatter for $\gamma \lesssim 0.5$, and are sensitive to the sign of $\gamma$. Solid lines show a kinetic theory calculation in the limit of large imbalance, and using the local reduced temperature at peak density [17].

Figure 4

Effect of interaction strength on spin transport. (a) LR parameter $\gamma$, (b) $D_0^{\perp }$, and (c) ${\lambda }_0$ as a function of $(k_{F}a{)}^{-1}$. The error bars represent fit uncertainties. For these data, $(T/T_F{)}_i=0.18(4)$ and $N=40(10)\times 10^3$, where uncertainty is due to number variation between runs. In the range $0<(k_{F}a{)}^{-1}\lesssim 1.3$ (indicated in gray) both free atoms and Feshbach dimers are present, as discussed in the text and in Fig. 5. Solid lines show a kinetic theory calculation [17] at $(T/T_F{)}_i=0.20$; the dotted line in (c) shows the weakly interacting limit ${\lambda }_0=[\pi /(2k_{F}a)-1{]}^{-1}$ for a balanced $T=0$ gas [16]. The inset to (c) shows ${\lambda }_0^{-1}$, and includes a calculation using the momentum averaged upper branch $T$ matrix (solid line) as well as ${\lambda }_0^{-1}=(4{\epsilon }_F/3n){\mathcal{T}}^{-1}(\overrightarrow{0},0)$. The sign change of ${\lambda }_0$ at $0.4\le (k_{F}a{)}^{-1}\le 1$ is a robust feature of theory, and is consistent with our data.

Figure 5

Presence of dimers above the Feshbach resonance. At the indicated initial $(k_{F}a{)}^{-1}$, a superposition is created with $\theta =\pi /2$ and held for 3 ms. The field is then swept to 200.0 G in 5 ms, which magnetoassociates some lower-branch pairs into dimers with a binding energy of $h\times 200\mathrm{kHz}$. Dimers are identified with their rf dissociation spectrum, using an $80\text{−}\mu \mathrm{s}$ pulse near the 46.85 MHz spin-flip resonance from the $|+z⟩$ state to a previously unoccupied Zeeman state [42]. Each plot shows the transfer rate versus rf frequency. For traces (a), (b), and (c), there is a clearly identified molecular feature above 47.0 MHz (spectral weight shaded in orange). However for traces (d), (e), and (f), the spectral weight above 47.0 MHz is insensitive to field, and consistent with the noise of the measurement.