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arXiv:1501.02677 · Physical Review A

Nonlinear spin diffusion and spin rotation in a trapped Fermi gas

Tilman Enss

Published Feb 17, 2015• 7 citations
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Abstract

Transverse spin diffusion in a polarized, interacting Fermi gas leads to the Leggett-Rice effect, where the spin current precesses around the local magnetization. With a spin-echo sequence both the transverse diffusivity and the spin-rotation parameter $\ensuremath{\gamma}$ are obtained; the sign of $\ensuremath{\gamma}$ reveals the repulsive or attractive character of the effective interaction. In a trapped Fermi gas the spin diffusion equations become nonlinear, and their numerical solution exhibits an inhomogeneous spin state even at the spin-echo time. While the microscopic diffusivity and $\ensuremath{\gamma}$ increase at weak coupling, their apparent values inferred from the trap-averaged magnetization saturate in agreement with a recent experiment for a dilute ultracold Fermi gas.

DOI
10.1103/physreva.91.023614

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Transverse spin diffusion in a polarized, interacting Fermi gas leads to the Leggett-Rice effect, where the spin current precesses around the local magnetization. With a spin-echo sequence both the transverse diffusivity and the spin-rotation parameter \ensuremathγ\ensuremath{\gamma}\ensuremathγ are obtained; the sign of \ensuremathγ\ensuremath{\gamma}\ensuremathγ reveals the repulsive or attractive character of the effective interaction. In a trapped Fermi gas the spin diffusion equations become nonlinear, and their numerical solution exhibits an inhomogeneous spin state even at the spin-echo time. While the microscopic diffusivity and \ensuremathγ\ensuremath{\gamma}\ensuremathγ increase at weak coupling, their apparent values inferred from the trap-averaged magnetization saturate in agreement with a recent experiment for a dilute ultracold Fermi gas.
We observe that the diffusive spin current in a strongly interacting degenerate Fermi gas of (40)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 γ=1.08(9) and a bare transverse spin diffusivity D(0)(⊥)=2.3(4)ℏ/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 γ=0 at unitarity, where two-body scattering is purely dissipative. We observe γ→0 as temperature is increased towards the Fermi temperature, consistent with calculations that show the degenerate Fermi sea restores a nonzero γ. Tuning the scattering length a, we find that a sign change in γ occurs in the range 0<(k(F)a)(-1)≲1.3, where k(F) is the Fermi momentum. We discuss how γ reveals the effective interaction strength of the gas, such that the sign change in γ indicates a switching of branch between a repulsive and an attractive Fermi gas.
We measure the transport properties of two-dimensional ultracold Fermi gases during transverse demagnetization in a magnetic field gradient. Using a phase-coherent spin-echo sequence, we are able to distinguish bare spin diffusion from the Leggett-Rice effect, in which demagnetization is slowed by the precession of a spin current around the local magnetization. When the two-dimensional scattering length is tuned to be comparable to the inverse Fermi wave vector k_{F}^{-1}, we find that the bare transverse spin diffusivity reaches a minimum of 1.7(6)ℏ/m, where m is the bare particle mass. The rate of demagnetization is also reflected in the growth rate of the s-wave contact, observed using time-resolved spectroscopy. The contact rises to 0.28(3)k_{F}^{2} per particle, which quantifies how scaling symmetry is broken by near-resonant interactions, unlike in unitary three-dimensional systems. Our observations support the conjecture that, in systems with strong scattering, the local relaxation rate is bounded from above by k_{B}T/ℏ.
We compute spin diffusion in a dilute Fermi gas at arbitrary temperature, polarization, and strong interaction in the normal phase using kinetic theory. While the longitudinal spin diffusivity ${D}_{\ensuremath{\parallel}}$ depends weakly on polarization and diverges for small temperatures, the transverse spin diffusivity ${D}_{\ensuremath{\perp}}$ has a strong polarization dependence and approaches a finite value for $T\ensuremath{\rightarrow}0$ in the Fermi liquid phase. For a 3D unitary Fermi gas at infinite scattering length, the diffusivities reach a minimum near the quantum limit of diffusion $\ensuremath{\hbar}/m$ in the quantum degenerate regime and are strongly suppressed by medium scattering, and we discuss the importance of the spin-rotation effect. In two dimensions, ${D}_{\ensuremath{\perp}}$ attains a minimum at strong coupling $\ensuremath{-}1\ensuremath{\lesssim}\mathrm{ln}({k}_{F}{a}_{2D})\ensuremath{\lesssim}1$ and reaches ${D}_{\ensuremath{\perp}}\ensuremath{\sim}0.2...\phantom{\rule{0.16em}{0ex}}0.3\phantom{\rule{0.16em}{0ex}}\ensuremath{\hbar}/m$ at large polarization. These values are consistent with recent measurements of two-dimensional ultracold atomic gases in the strong coupling regime.
We have studied NMR spin echoes in 3He^{3}\mathrm{He}3He liquid in a magnetic field B0{\mathit{B}}_{0}B0​=8 T for temperatures T\ensuremath{\ge}4.5 mK. The data are analyzed in terms of the Leggett-Rice effect to obtain the transverse spin-diffusion coefficient ${\mathit{D}}_{\mathrm{\ensuremath{\perp}}}$(T). For T20 mK we find that ${\mathit{D}}_{\mathrm{\ensuremath{\perp}}}$ is less than measured in earlier experiments at lower B0{\mathit{B}}_{0}B0​. This phenomenon has been predicted for degenerate Fermi systems, due to the phase space created for quasiparticle scattering by spin polarization. The effect we have measured is more pronounced than suggested by scaling a dilute-gas theory result by a Fermi-liquid factor.
Recent experiments in a mixture of two hyperfine states of trapped Bose gases show behavior analogous to a spin-12\frac{1}{2}21​ system, including transverse spin waves and other familiar Leggett-Rice-type effects. We have derived the kinetic equations applicable to these systems, including the spin dependence of interparticle interactions in the collision integral, and have solved for spin-wave frequencies and longitudinal and transverse diffusion constants in the Boltzmann limit. We find that, while the transverse and longitudinal collision times for trapped Fermi gases are identical, the Bose gas shows unusual diffusion anisotropy in both dipole and quadrupole modes. Moreover, the lack of spin isotropy in the interactions leads to the nonconservation of transverse spin, which in turn has significant effects on the hydrodynamic modes.
We solve a recently derived kinetic equation for a dilute quantum system at arbitrary degeneracy and polarization, M. From this equation we develop a spin hydrodynamic equation that allows the generalization of the treatment of spin waves and the Leggett-Rice effect to polarized degenerate Fermi gases. We find a transverse spin-diffusion collision time ${\ensuremath{\tau}}_{\ensuremath{\perp}}$ that is often shorter than the corresponding longitudinal collision time ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\parallel}}}$ that is often shorter than the corresponding longitudinal collision time ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$. As T\ensuremath{\rightarrow}0, ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\parallel}}}$ approaches T-independent value, in constrast to ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\sim}1/T2{\mathit{T}}^{2}T2. Spin waves would thus continue to be damped even to T=0 K.