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arXiv:1207.3103 · Physical Review Letters

Quantum Mechanical Limitations to Spin Diffusion in the Unitary Fermi Gas

Tilman Enss, R. Haussmann

Published Nov 9, 2012• 77 citations

Abstract

We compute spin transport in the unitary Fermi gas using the strong-coupling Luttinger-Ward theory. In the quantum degenerate regime the spin diffusivity attains a minimum value of ${D}_{s}\ensuremath{\simeq}1.3\ensuremath{\hbar}/m$ approaching the quantum limit of diffusion for a particle of mass $m$. Conversely, the spin drag rate reaches a maximum value of ${\ensuremath{\Gamma}}_{\mathrm{sd}}\ensuremath{\simeq}1.2{k}_{B}{T}_{F}/\ensuremath{\hbar}$ in terms of the Fermi temperature ${T}_{F}$. The frequency-dependent spin conductivity ${\ensuremath{\sigma}}_{s}(\ensuremath{\omega})$ exhibits a broad Drude peak, with spectral weight transferred to a universal high-frequency tail ${\ensuremath{\sigma}}_{s}(\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\infty})={\ensuremath{\hbar}}^{1/2}C/3\ensuremath{\pi}(m\ensuremath{\omega}{)}^{3/2}$ proportional to the Tan contact density $C$. For the spin susceptibility ${\ensuremath{\chi}}_{s}(T)$ we find no downturn in the normal phase.

DOI: 10.1103/physrevlett.109.195303

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We compute spin transport in the unitary Fermi gas using the strong-coupling Luttinger-Ward theory. In the quantum degenerate regime the spin diffusivity attains a minimum value of

{D}_{s}\ensuremath{\simeq}1.3\ensuremath{\hbar}/m

approaching the quantum limit of diffusion for a particle of mass

m

. Conversely, the spin drag rate reaches a maximum value of

{\ensuremath{\Gamma}}_{\mathrm{sd}}\ensuremath{\simeq}1.2{k}_{B}{T}_{F}/\ensuremath{\hbar}

in terms of the Fermi temperature

{T}_{F}

. The frequency-dependent spin conductivity

{\ensuremath{\sigma}}_{s}(\ensuremath{\omega})

exhibits a broad Drude peak, with spectral weight transferred to a universal high-frequency tail

{\ensuremath{\sigma}}_{s}(\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\infty})={\ensuremath{\hbar}}^{1/2}C/3\ensuremath{\pi}(m\ensuremath{\omega}{)}^{3/2}

proportional to the Tan contact density

C

. For the spin susceptibility

{\ensuremath{\chi}}_{s}(T)

we find no downturn in the normal phase.

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.

The thermodynamic and transport properties of the unitary Fermi gas at finite temperature

T

are governed by a quantum critical point at

T=0

and zero density. We compute the universal shear viscosity to entropy ratio $\ensuremath{\eta}/s$ in the high-temperature quantum critical regime $T\ensuremath{\gg}|\ensuremath{\mu}|$ and find that this strongly coupled quantum fluid comes close to perfect fluidity $\ensuremath{\eta}/s=\ensuremath{\hbar}/(4\ensuremath{\pi}{k}_{B})$. Using a controlled large-

N

expansion, we show that already at the first nontrivial order the equation of state and the Tan contact density

C

agree well with the most recent experimental measurements and theoretical Luttinger-Ward and bold diagrammatic Monte Carlo calculations.

Understanding the quantum dynamics of strongly interacting fermions is a problem relevant to diverse forms of matter, including high-temperature superconductors, neutron stars, and quark-gluon plasma. An appealing benchmark is offered by cold atomic gases in the unitary limit of strong interactions. Here, we study the dynamics of a transversely magnetized unitary Fermi gas in an inhomogeneous magnetic field. We observe the demagnetization of the gas, caused by diffusive spin transport. At low temperatures, the diffusion constant saturates to the conjectured quantum-mechanical lower bound ≃ ħ/m, where m is the particle mass. The development of pair correlations, indicating the transformation of the initially noninteracting gas toward a unitary spin mixture, is observed by measuring Tan's contact parameter.

We examine spin diffusion in a two-component homogeneous Fermi gas in the normal phase. Using a variational approach, analytical results are presented for the spin diffusion coefficient and the related spin relaxation time as a function of temperature and interaction strength. For low temperatures, strong correlation effects are included through the Landau parameters, which we extract from Monte Carlo results. We show that the spin diffusion coefficient has a minimum for a temperature somewhat below the Fermi temperature with a value that approaches the quantum limit ∼ℏ/ m in the unitarity regime, where m is the particle mass. Finally, we derive a value for the low-temperature shear viscosity in the normal phase from the Landau parameters.

This thesis investigates the dynamics of a degenerate Fermi gas in the unitary limit of strong interactions, in particular, the dynamics of a transversely magnetized unitary Fermi gas in an inhomogeneous magnetic field. Short-range interactions between ultracold fermionic potassium atoms are tuned to their quantum mechanical limit by employing a Feshbach resonance. Spin-echo measurements of magnetization dynamics show that demagnetization is caused by diffusive spin transport with a diffusion constant that saturates at low temperatures to the conjectured quantum-mechanical lower bound ~/m, where m is the particle mass. A prediction for the increase in temperature due to demagnetization based on universal thermodynamics is compared to measured momentum distributions. The development of pair correlations is observed by time-resolved radiofrequency spectroscopy of Tan's contact parameter.