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arXiv:1308.3162 · Physical Review Letters

Observation of Conformal Symmetry Breaking and Scale Invariance in Expanding Fermi Gases

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

Published Jan 29, 2014• 63 citations
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Abstract

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.

DOI
10.1103/physrevlett.112.040405

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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.
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 study the hydrodynamic expansion of a rotating strongly interacting Fermi gas by releasing a cigar-shaped cloud with a known angular momentum from an optical trap. As the aspect ratio of the expanding cloud approaches unity, the angular velocity increases, indicating quenching of the moment of inertia I to as low as 0.05 of the rigid body value I(rig). Remarkably, we observe this behavior in both the superfluid and collisional normal fluid regimes, which obey nearly identical zero-viscosity irrotational hydrodynamics. We attribute irrotational flow in the normal fluid to a decay of the rotational part of the stream velocity during expansion, which occurs when the shear viscosity is negligible. Using conservation of angular momentum, we directly observe a fundamental result of irrotational hydrodynamics, I/I(rig) = delta2, where delta is the deformation parameter of the cloud.
We investigate collective excitations of a harmonically trapped two-dimensional Fermi gas from the collisionless (zero sound) to the hydrodynamic (first sound) regime. The breathing mode, which is sensitive to the equation of state, is observed with an undamped amplitude at a frequency 2 times the dipole mode frequency for a large range of interaction strengths and different temperatures. This provides evidence for a dynamical SO(2,1) scaling symmetry of the two-dimensional Fermi gas. Moreover, we investigate the quadrupole mode to measure the shear viscosity of the two-dimensional gas and study its temperature dependence.
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 unitary Fermi gas in an isotropic harmonic trap is predicted to show scale and conformal symmetry that have important consequences in its thermodynamic and dynamical properties. By experimentally realizing a unitary Fermi gas in an isotropic harmonic trap, we demonstrate its universal expansion dynamics along each direction and at different temperatures. We show that as a consequence of SO(2,1) symmetry, the measured release energy is equal to that of the trapping energy. We further observe the breathing mode with an oscillation frequency twice the trapping frequency and a small damping rate, providing the evidence of SO(2,1) symmetry. In addition, away from resonance when scale invariance is broken, we determine the effective exponent γ that relates the chemical potential and average density along the BEC-BCS crossover, which qualitatively agrees with the mean field predictions. This Letter opens the possibility of studying nonequilibrium dynamics in a conformal invariant system in the future.