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Quantum Transport and Scale Invariance in Expanding Fermi Gases

Ethan Elliott

Published Jan 1, 2014• 1 citation
OpenAlex

Abstract

This dissertation describes the first experimental measurement of the energy and interaction dependent shear viscosity $\\eta$ and bulk viscosity $\\zeta$ 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 $\\langle{\\mathbf{r}}^2\\rangle=\\langle x^2+y^2+z^2\\rangle$ expands ballistically (like a noninteracting gas) and the energy-averaged bulk viscosity is consistent with zero, $0.00(0.04)\\,\\hbar\\,n$, 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 $\\langle{\\mathbf{r}}^2\\rangle$ deviates from ballistic flow. We find that $\\eta$ has both a quadratic and a linear dependence on the interaction strength $1/({k_{FI}a})$, where $a$ is the s-wave scattering length and $k_{FI}$ 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 $1/(k_{FI}a) = 0.2$.

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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.
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.
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 measure the shear viscosity in a two-component Fermi gas of atoms, tuned to a broad s-wave collisional (Feshbach) resonance. At resonance, the atoms strongly interact and exhibit universal behavior, where the equilibrium thermodynamic properties and transport coefficients are universal functions of density n and temperature T . We present a new calibration of the temperature as a function of global energy, which is directly measured from the cloud profiles. Using the calibration, the trap-averaged shear viscosity in units of ℏ n is determined as a function of the reduced temperature at the trap center, from nearly the ground state to the unitary two-body regime. Low-temperature data are obtained from the damping rate of the radial breathing mode, whereas high-temperature data are obtained from hydrodynamic expansion measurements. We also show that the best fit to the high-temperature expansion data is obtained for a vanishing bulk viscosity. The measured trap-averaged entropy per particle and shear viscosity are used to estimate the ratio of shear viscosity to entropy density, which is compared with that conjectured for a perfect fluid.
We measure the shear viscosity in a two-component Fermi gas of atoms, tuned to a broad s-wave collisional (Feshbach) resonance. At resonance, the atoms strongly interact and exhibit universal behavior, where the equilibrium thermodynamic properties and transport coefficients are universal functions of density n and temperature T. We present a new calibration of the temperature as a function of global energy, which is directly measured from the cloud profiles. Using the calibration, the trap-averaged shear viscosity in units of ℏn is determined as a function of the reduced temperature at the trap center, from nearly the ground state to the unitary two-body regime. Low-temperature data are obtained from the damping rate of the radial breathing mode, whereas high-temperature data are obtained from hydrodynamic expansion measurements. We also show that the best fit to the high-temperature expansion data is obtained for a vanishing bulk viscosity. The measured trap-averaged entropy per particle and shear viscosity are used to estimate the ratio of shear viscosity to entropy density, which is compared with that conjectured for a perfect fluid.
We measure the static shear viscosity η in a two-component Fermi gas near a broad collisional (Feshbach) resonance, as a function of interaction strength and energy. We find that η has both a quadratic and a linear dependence on the interaction strength 1/(k(FI)a), where a is the s-wave scattering length and k(FI) is the Fermi wave vector for an ideal gas at the trap center. For energies above the superfluid transition, the minimum in η as a function of interaction strength is significantly shifted toward the BEC side of resonance, to 1/(k(FI)a)≃0.25.
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.