Universal Quantum Viscosity in a Unitary Fermi Gas

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Science  07 Jan 2011:
Vol. 331, Issue 6013, pp. 58-61
DOI: 10.1126/science.1195219


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 pT3/ħ2, where pT 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 T3/2 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.

Ultracold, strongly interacting Fermi gases are of broad interest because they provide a tunable tabletop paradigm for strongly interacting systems, ranging from high-temperature superconductors to nuclear matter. First observed in 2002, quantum degenerate, strongly interacting Fermi gases are being widely studied (14). To obtain strong interactions (characterized by a divergent s-wave scattering length), a bias magnetic field is used to tune the gas to a broad collisional (Feshbach) resonance, for which the range of the collision potential is small compared with the interparticle spacing. In this so-called unitary regime, the properties of the gas are universal functions of the density n and temperature T. The universal behavior of the equilibrium thermodynamic properties has been studied in detail (511), whereas the measurement of universal transport coefficients presents new challenges.

The measurement of the viscosity is of particular interest in the context of a recent conjecture, derived using string theory methods, that defines a perfect normal fluid (12). An example of a nearly perfect fluid is the quark-gluon plasma produced in gold ion collisions, which exhibits almost perfect frictionless flow and is thought to be a good approximation to the state of matter that existed microseconds after the Big Bang (13). The conjecture states that the ratio of the shear viscosity η to the entropy density s has a universal minimum, η/sħ/(4πkB), where ħ is Planck’s constant h divided by 2π and kB is the Boltzmann constant. This ratio is experimentally accessible in a trapped unitary Fermi gas, in which the entropy has been measured both globally (6, 9) and locally (10, 11) and the viscosity can be determined from hydrodynamic experiments (1417), so that the predicted minimum ratio can be directly compared with that from Fermi gas experiments (16, 17).

In a Fermi gas, the η/s ratio for the normal fluid is expected to reach a minimum just above the superfluid transition temperature (16). This can be understood by using dimensional analysis. Shear viscosity has units of momentum per area. For a unitary gas, the natural momentum is the relative momentum ħ k of a colliding pair of particles, whereas the natural area is the resonant s-wave collision cross section, 4π/k2 (18). Thus, η ∝ ħ k3. At temperatures well below the Fermi temperature at which degeneracy occurs, the Fermi momentum sets the scale so that k ≅ 1/L, where L is the interparticle spacing. Then, η ∝ ħ /L3, and η ∝ ħ n. For a normal fluid above the critical temperature, the scale of entropy density sn kB, so that η/sħ /kB. For much higher temperatures above the Fermi temperature, one expects that ħ k is comparable with the thermal momentum pT=2mkBT, giving the scale η ∝ pT3/ħ2T3/2/ħ2.

To properly measure the shear viscosity with high precision over a wide temperature range, we used universal hydrodynamic equations, which contain both the friction force and the heating rate, to extract the viscosity from two experiments, one for each of two temperature ranges. For measurement at high temperatures, we observed the expansion dynamics of a unitary Fermi gas after release from a deep optical trap and demonstrated the predicted universal T3/2 temperature scaling. For measurement at low temperatures, we used the damping rate of the radial breathing mode, using the raw cloud profiles from our previous work (19). The smooth joining of the data from the two measurement methods when heating is included (20), and the discontinuity of the data when heating is excluded, demonstrates the importance of including the heating as well as the friction force in the universal hydrodynamic analysis.

The experiments employ a 50-50 mixture of the two lowest hyperfine states of 6Li, which was magnetically tuned to a broad Feshbach resonance and cooled by means of evaporation in the optical trap. The initial energy per particle E is measured from the trapped cloud profile (20).

In the high-temperature regime, the total energy of the gas E is larger than 2EF, well above the critical energy Ec < 0.8EF for the superfluid transition (911). In this case, the density profile is well fit by a Gaussian, n(x,y,z,t) = n0(t) exp(−x2x2y2y2z2z2), where σi(t) is a time-dependent width, n0(t)=N/(π3/2σxσyσz) is the central density, and N is the total number of atoms.

The aspect ratio σx(t)/σz(t) was measured as a function of time after release so as to characterize the hydrodynamics, for different energies E between 2.3EF and 4.6EF (Fig. 1). We also took expansion data at one low-energy point E = 0.6EF, where the viscosity is small as compared with that obtained at higher temperatures and the density profile is approximately a zero-temperature Thomas-Fermi distribution. The black curve in Fig. 1 shows the fit for zero viscosity and no free parameters. To obtain a high signal-to-background ratio, we measured the aspect ratio only up to 1.4. For comparison, the green dashed curve in Fig. 1 shows the prediction for a ballistic gas.

Fig. 1

Anisotropic expansion. (A) Cloud absorption images for 0.2, 0.3, 0.6, 0.9, and 1.2 ms expansion time; E = 2.3EF. (B) Aspect ratio versus time. The expansion rate decreases at higher energy as the viscosity increases. Solid curves indicate hydrodynamic theory, with the viscosity as the fit parameter. Error bars denote statistical fluctuations in the aspect ratio.

We determined the shear viscosity η by using a hydrodynamic description of the velocity field v(x,t) in terms of the scalar pressure and the shear viscosity pressure tensor, m(t+v·)vi=fi+jj(ησij)n(1)where f = −∇ P/n is the force per particle arising from the scalar pressure P and m is the atom mass. For a unitary gas, the bulk viscosity is predicted to vanish in the normal fluid (21, 22), so we did not include it in the analysis for the expansion. The second term on the right describes the friction forces arising from the shear viscosity, where σij = ∂vi/∂xj + ∂vj/∂xi − 2δij∇ · v/3 is symmetric and traceless.

For a unitary gas, the evolution equation for the pressure takes a simple form because P = 2E/3 (23, 24), where E is the local energy density (sum of the kinetic and interaction energy). Then, energy conservation and Eq. 1 implies (∂t + v · ∇ + 5∇ · v/3)P = 2 · q˙/3. Here, the heating rate per unit volume times q˙ = ησij2/2 arises from friction from the relative motion of neighboring volume elements. To express this in terms of the force per particle (fi), we differentiated this equation for P with respect to xi and used the continuity equation for the density to obtain(t+v·+23·v)fi+j(ivj)fj53(iv)Pn=23iq˙n(2)Force balance in the trapping potential Utrap(x), just before release of the cloud, determines the initial condition fi(0) = ∂iUtrap(x).

These hydrodynamic equations include both the force and the heating arising from viscosity. The solution is greatly simplified when the cloud is released from a deep, nearly harmonic trapping potential Utrap because fi(0) is then linear in the spatial coordinate. If we neglect viscosity, the force per particle and hence the velocity field remain linear functions of the spatial coordinates as the cloud expands. Thus, ∂i(∇ · v) = 0, and the pressure P does not appear in Eq. 2. Through numerical integration (25), we found that nonlinearities in the velocity field are very small, even if the viscosity is not zero, because dissipative forces tend to restore a linear flow profile. Hence, the evolution Eqs. 1 and 2 are only weakly dependent on the precise initial spatial profile of P and independent of the detailed thermodynamic properties.

We therefore assumed that the velocity field is exactly linear in the spatial coordinates. We took fi = ai(t)xi and σi(t) = bi(ti(0); the density changes by a scale transformation (26), where current conservation then requires vi = xi bi(t)/bi(t).

In general, the viscosity takes the universal form η = α(θ)ħn, where θ is the local reduced temperature and η → 0 in the low-density region of the cloud (20, 27). Using the measured trap frequencies, and Eqs. 1 and 2, the aspect ratio data are fit to determine the trap-averaged viscosity parameter, α¯ = (1/)∫d3x η(x,t), which arises naturally independent of the spatial profile of and is equivalent to assuming η. Because θ has a zero convective derivative everywhere (in the zeroth-order adiabatic approximation) and the number of atoms in a volume element is conserved along a stream tube, α¯ is a constant that can be compared with predictions for the trapped cloud before release.

As shown in Fig. 1, the expansion data are very well fit over the range of energies studied, using α¯ as the only free parameter. We found that the friction force produces a curvature that matches the aspect ratio–versus-time data, whereas the indirect effect of heating is important in increasing the outward force, which increases the fitted α¯ by a factor of ≅2, as compared with that obtained when heating is omitted (20).

For measurements at low temperatures, where the viscosity is small, we determined α¯ from the damping rate of the radial breathing mode (19). For the breathing mode, the cloud radii change by a scale transformation of the form bi = 1 + ε˙i, with εi << 1, and the heating rate in Eq. 2 is ∝ ε˙i2, which is negligible. Hence, the force per particle evolves adiabatically. Adding the trapping force to Eq. 1, one obtains the damping rate 1/τ = ħ α¯/(3mx2) (20, 28).

The fitted viscosity coefficients α¯ for the entire energy range are shown in Fig. 2, which can be used to test predictions (2931). Despite the large values of α¯ at the higher energies, the viscosity causes only a moderate perturbation to the adiabatic expansion, as shown by the expansion data and the fits in Fig. 1. The breathing mode data and expansion data smoothly join, provided that the heating rate is included in the analysis. In contrast, omitting the heating rate produces a discontinuity between the high- and low-temperature viscosity data (20). The agreement between these very different measurements when heating is included shows that hydrodynamics in the universal regime is well described by Eqs. 1 and 2.

Fig. 2

Trap-averaged viscosity coefficient Embedded Image = ∫d3x η/(ħN) versus initial energy per atom. Blue circles indicate breathing-mode measurements; red squares indicate anisotropic expansion measurements. Bars denote statistical error arising from the uncertainty in E and the cloud dimensions. (Inset) Embedded Image versus reduced temperature θ0 at the trap center before release of the cloud. The blue curve shows the fit α0 = α3/2 θ03/2, demonstrating the predicted universal high-temperature scaling. Bars denote statistical error arising from the uncertainty in θ0 and Embedded Image. A 3% systematic uncertainty in EF and 7% in θ0 arises from the systematic uncertainty in the absolute atom number (20).

To test the prediction of the T3/2 temperature scaling in the high-temperature regime, we assumed that η relaxes to the equilibrium value in the center of the trap but vanishes in the low-density region so that α¯ is well defined. This behavior is predicted by kinetic theory (27). We expect that α¯ ≅ α0, where η0 = α0 ħ n0 is the viscosity at the trap center before release. At high temperatures (15),α0=α3/2θ03/2(3)where α3/2 is a universal coefficient. Because θ has a zero convective derivative everywhere (in the zeroth-order adiabatic approximation), θ0 at the trap center has a zero time derivative, and α0 is therefore constant, as is α¯.

The inset in Fig. 2 shows the high-temperature (expansion) data for α¯ versus the initial reduced temperature at the trap center, θ0. Here, θ0 = T0/TF(n0) = (T0/TFI)(nI/n0)2/3. The local Fermi temperature TF(n0) = ħ2(3π2n0)2/3/(2mkB), and TFI = EF /kB = TF (nI) is the ideal gas Fermi temperature at the trap center. nI is the ideal gas central density for a zero-temperature Thomas-Fermi distribution. We used (nI/n0)2/3 = 4(σ2z2Fz)/π1/3 and obtained the initial T0/TFI from the cloud profile (20).

The excellent fit of Eq. 3 to the data (Fig. 2, inset) demonstrates that at high temperature, the viscosity coefficient very well obeys the θ03/2 scaling, which is in agreement with predictions (15). Eq. 3 predicts that α0 scales nearly as E3 because θ0T0/n02/3E2. This explains the factor of ≅10 increase in the viscosity coefficients as the initial energy is increased from E = 2.3EF to E = 4.6EF.

A precise comparison between the viscosity data and theory requires calculation of the trap-average α¯ from the local shear viscosity, where the relation is tightly constrained by the observed T3/2 scaling. Our simple approximation α¯ ≅ α0 yields α3/2 = 3.4(0.03), where 0.03 is the statistical error from the fit. A better estimate based on a relaxation model (29) shows that α¯ = 1.3 α0 at high T, yielding α3/2 = 2.6. At sufficiently high temperature, the mean free path becomes longer than the interparticle spacing because the unitary collision cross section decreases with increasing energy. In this limit, a two-body Boltzmann equation description of the viscosity is valid. For a Fermi gas in a 50-50 mixture of two spin states, a variational calculation (15) yields α3/2 = 45π3/2/(642) = 2.77, which is in reasonable agreement with the fitted values.

Lastly, Fig. 3 shows an estimate of the ratio of η/s = αħn/s = (ħ/kB)α¯/(s/nkB) ≅ (ħ/kB) α/S, where S is the average entropy per particle of the trapped gas in units of kB. We obtain S in the low-temperature regime from (9), which joins smoothly to the second virial coefficient approximation for S in the high-temperature regime (20). The Fig. 3 inset shows the low-temperature behavior, which is about five times the string theory limit (Fig. 3, inset, red dashed line) near the critical energy Ec/EF = 0.7−0.8 (9, 20). The apparent decrease of the η/s ratio as the energy approaches the ground state 0.48EF (9) does not require that the local ratio → 0 as T → 0 because contributions from the cloud edges significantly increase S as compared with the local s at the center.

Fig. 3

Estimated ratio of the shear viscosity to the entropy density. Blue circles indicate breathing-mode measurements; red squares indicate anisotropic expansion measurements. (Inset) The red dashed line denotes the string theory limit. Bars denote statistical error arising from the uncertainty in E, Embedded Image, and S (20).

Supporting Online Material

Materials and Methods

Figs. S1 and S2


References and Notes

  1. The experiments were performed far from p-wave Feshbach resonances. The relevant threshold energy for p-wave scattering was then comparable with the barrier height. Using the known C6 coefficients, the barrier height for 40K is 280 μK, whereas for 6Li the barrier height is 8 mK. Hence, for temperatures in the μK range as used in the experiments, p-wave scattering is negligible, and s-wave scattering dominates.
  2. Materials and methods are available as supporting material on Science Online.
  3. We give the damping rate 1/τ for a cylindrically symmetric cigar-shaped trap. For δ ≡ (ωx – ωy)/ωxωy << 1, with ωxy the transverse trap frequencies, 1/τ contains an additional factor 1 – δ.
  4. This research is supported by the Physics Divisions of NSF, the Army Research Office, the Air Force Office Office of Sponsored Research, and the Division of Materials Science and Engineering, the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy. J.E.T. and T.S. thank the ExtreMe Matter Institute (EMMI) for hospitality.
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