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Scale Invariance in Expanding Fermi Gases

Ethan Elliott, James A. Joseph, John E. Thomas

Published Aug 14, 2013• 0 citations
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Abstract

We precisely test scale invariance in the hydrodynamic expansion of a Fermi gas of atoms with resonant interactions, using a model-independent method. We observe that the three-dimensional mean square cloud radius expands ballistically and find that the bulk viscosity is consistent with zero, as predicted for a scale-invariant system. At resonance, the expanding cloud obeys the universal equation of state for a non-relativistic gas, $p-{2}{3}\, {\cal E}=0$, where $p$ is the pressure and ${\cal E}$ is the energy density. Tuning away from resonance, we show that the observed deviation from scale invariant flow can be explained by the change in the equation of state $p-{2}{3}\, {\cal E}\neq 0$ (conformal symmetry breaking parameter) and a finite bulk viscosity, which is compared to predictions.

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Physical Review Letters•Sep 19, 2013•42 citations
We precisely test scale invariance in the hydrodynamic expansion of a Fermi gas of atoms with resonant interactions, using a model-independent method. We observe that the three-dimensional mean square cloud radius expands ballistically and find that the bulk viscosity is consistent with zero, as predicted for a scale-invariant system. At resonance, the expanding cloud obeys the universal equation of state for a non-relativistic gas, p−23 E=0p-{2}{3}\, {\cal E}=0p−23E=0, where ppp is the pressure and E{\cal E}E is the energy density. Tuning away from resonance, we show that the observed deviation from scale invariant flow can be explained by the change in the equation of state p−23 E≠0p-{2}{3}\, {\cal E}\neq 0p−23E=0 (conformal symmetry breaking parameter) and a finite bulk viscosity, which is compared to predictions.
We precisely test scale invariance and examine local thermal equilibrium in the hydrodynamic expansion of a Fermi gas of atoms as a function of interaction strength. After release from an anisotropic optical trap, we observe that a resonantly interacting gas obeys scale-invariant hydrodynamics, where the mean square cloud size <r2>=<x2+y2+z2> expands ballistically (like a noninteracting gas) and the energy-averaged bulk viscosity is consistent with zero, 0.00(0.04)ℏn, with n the density. In contrast, the aspect ratios of the cloud exhibit anisotropic "elliptic" flow with an energy-dependent shear viscosity. Tuning away from resonance, we observe conformal symmetry breaking, where <r2> deviates from ballistic flow.
This dissertation describes the first experimental measurement of the energy and interaction dependent shear viscosity eta\\etaeta and bulk viscosity zeta\\zetazeta in the hydrodynamic expansion of a two-component Fermi gas near a broad collisional (Feshbach) resonance. This expansion also provides a precise test of scale invariance and an examination of local thermal equilibrium as a function of interaction strength. After release from an anisotropic optical trap, we observe that a resonantly interacting gas obeys scale-invariant hydrodynamics, where the mean square cloud size langlemathbfr2rangle=langlex2+y2+z2rangle\\langle{\\mathbf{r}}^2\\rangle=\\langle x^2+y^2+z^2\\ranglelanglemathbfr2rangle=langlex2+y2+z2rangle expands ballistically (like a noninteracting gas) and the energy-averaged bulk viscosity is consistent with zero, 0.00(0.04),hbar,n0.00(0.04)\\,\\hbar\\,n0.00(0.04),hbar,n, with nnn the density. In contrast, the aspect ratios of the cloud exhibit anisotropic ``elliptic" flow with an energy-dependent shear viscosity. Tuning away from resonance, we observe conformal symmetry breaking, where langlemathbfr2rangle\\langle{\\mathbf{r}}^2\\ranglelanglemathbfr2rangle deviates from ballistic flow. We find that eta\\etaeta has both a quadratic and a linear dependence on the interaction strength 1/(kFIa)1/({k_{FI}a})1/(kFI​a), where aaa is the s-wave scattering length and kFIk_{FI}kFI​ is the Fermi wave vector for an ideal gas at the trap center. At low energy, the minimum is less than the resonant value and is significantly shifted toward the BEC side of resonance, to 1/(kFIa)=0.21/(k_{FI}a) = 0.21/(kFI​a)=0.2.
A Fermi gas of atoms with resonant interactions is predicted to obey universal hydrodynamics, where the shear viscosity and other transport coefficients are universal functions of the density and temperature. At low temperatures, the viscosity has a universal quantum scale ℏn\hbar nℏn where nnn is the density, while at high temperatures the natural scale is pT3/ℏ2p_T^3/\hbar^2pT3​/ℏ2 where pTp_TpT​ is the thermal momentum. We employ breathing mode damping to measure the shear viscosity at low temperature. At high temperature TTT, we employ anisotropic expansion of the cloud to find the viscosity, which exhibits precise T3/2T^{3/2}T3/2 scaling. In both experiments, universal hydrodynamic equations including friction and heating are used to extract the viscosity. We estimate the ratio of the shear viscosity to the entropy density and compare to that of a perfect fluid.
The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio (P−23E)/P(P-\frac{2}{3}{\cal E})/P(P−32​E)/P, where PPP is the pressure and E{\cal E}E is the energy density. In the high temperature limit this ratio scales as zλa\frac{zλ}{a}azλ​, where zzz is the fugacity, λλλ is the thermal wave length, and aaa is the scattering length. We show that the bulk viscosity ζζζ scales as the second power of this parameter, ζ∼(zλa)2λ−3ζ\sim (\frac{zλ}{a})^2 λ^{-3}ζ∼(azλ​)2λ−3.
A Fermi gas of atoms with resonant interactions is predicted to obey universal hydrodynamics, in which the shear viscosity and other transport coefficients are universal functions of the density and temperature. At low temperatures, the viscosity has a universal quantum scale ħ n, where n is the density and ħ is Planck's constant h divided by 2π, whereas at high temperatures the natural scale is p(T)(3)/ħ(2), where p(T) is the thermal momentum. We used breathing mode damping to measure the shear viscosity at low temperature. At high temperature T, we used anisotropic expansion of the cloud to find the viscosity, which exhibits precise T(3/2) scaling. In both experiments, universal hydrodynamic equations including friction and heating were used to extract the viscosity. We estimate the ratio of the shear viscosity to the entropy density and compare it with that of a perfect fluid.
The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute, in the high temperature limit, the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio $(P\ensuremath{-}2/3\mathcal{E})/P$, where PPP is the pressure and E\mathcal{E}E is the energy density. At high temperature this ratio scales as $z\ensuremath{\lambda}/a$, where zzz is the fugacity, $\ensuremath{\lambda}$ is the thermal wavelength, and aaa is the scattering length. We show that the bulk viscosity $\ensuremath{\zeta}$ scales as the second power of this parameter, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.