Universal Quantum Viscosity in a Unitary Fermi Gas

Transcription

1 Universal Quantum Viscosity in a Unitary Fermi Gas Chenglin Cao Duke University July 5 th, 011 Advisor: Prof. John E Thomas Dr.Ilya Arakelyan Dr.James Joseph Dr.Haibin Wu Yingyi Zhang Ethan Elliott Willie Ong Chingyun Cheng Arun Jagannathan

2 Strongly Interacting Fermi Gases Similar Elliptic Flow 6 Lithium gas T = 10-7 K O Hara et al.,science (00) Quark-gluon plasma T = 10 1 K Photo courtesy: Brookhaven National Lab

3 String Theory Conjecture KSS Bound Shear Viscosity(Hydrodynamics) η s 1 π k 4 B Lower bound = perfect fluid Entropy Density(Thermodynamics) Kovtun, Son & Starinets, PRL(005)

4 Optically Trapped Fermi Gas 1 1 =,1, 0 = electron spin,nuclear spin

5 Feshbach Resonance Tunable interactions: nature (attractive vs. repulsive) and strength of interactions controlled by Feshbach resonance Strong interactions generated using formula in PRL (005)

6 Universality Near Feshbach resonance, a >> L >> R S-wave cross section: σ = 4 π /k The system is universal, i.e., the interparticle spacing L is the only length scale: 1) No dependence on microscopic details ) All local properties depend only on n,t

7 Quantum Shear Viscosity Shear forces d v F A = η d v Viscosity scale η = p σ p = k 4π σ = k 3 η = k Quantum scale requires Planck s constant

8 Quantum Viscosity at Low and High Temperature 3 η = k Low Temperature T T F High Temperature T T F k kf 1/ L k k Thermal mkb T / η n η T 3/ / Entropy density scale: s nk B Low temperature: η / s / k B String theory limit

9 Universal Shear Viscosity η (,) xt = αθ () nxt (,) High Temperature: Elliptical Flow Low Temperature: Breathing Mode

10 Universal Shear Viscosity η (,) xt = αθ () nxt (,) High Temperature: Elliptical Flow Low Temperature: Breathing Mode

11 MOT and FORT Atoms pre-cooled in a magneto-optical trap (MOT) to 150 μk Atoms loaded into far-off-resonance-trap (FORT) Atoms evaporatively cooled down to the desired temperature Atoms anisotropically expanded after being released from FORT

12 Elliptical Flow nxyzt n x y z (,,, ) = 0 exp( / σ x / y / σz) σ x σ x AspectRatio = σ z σ z

13 Hydrodynamic Theory Force Equation ' j( ησ ij + ςσ ij) t i i i trap j n m( + v ) v = f + U Force arising from the scale pressure Shear Viscosity σ = v + v δ v 3 ij i j j i ij f i ip = n Bulk Viscosity σ ' ij = δ v ij

14 Hydrodynamic Theory Energy Conservation ( 5 ) t + v + v = q Heating rate per unit volume 3 1 q = η σ + ς( v) ij Universal pressure P = ε Ho, PRL ε ij ( ) t + v + v = q 5 P 3 3

15 Hydrodynamic Theory Scale Transformation nxyzt (,,, ) = v i = b i xi b i x y z n(,, ) b x b y b z b () t b () t b () t x y z f = a () t mω x i i i i b(0) = 1, b (0) = 0, a (0) = 1 i i i

16 Hydrodynamic Theory b b a + a + a = i j i i i σ ij bi 3 j bj 3 mωi xi b ( ) ij 0 i t b α = ( a 1 ) ω b m x b t α σ i i trap i ii i i () 0 i b(0) = 1, b (0) = 0, a (0) = 1 i i i Trap-averaged viscosity coefficient α d x (,) x t d x ( )(,) n x t N η = N α θ The viscosity must vanish at the cloud edges

17 Energy Determination The initial energy per particle E is determined by exploiting the virial theorem E = U = m z z trap 3 ωz [1 κ / σfz ] σ Fz = EF /( mω z) κ = 15 E / (4 U ) F 0 Thomas (005) Castin (004) Werner and Castin (006) Son (007) z From the fit

18 Elliptical Flow

19 Universal High Temperature Scaling α = α θ 3/ 0 3/ 0 α 3/ = 3.4(0.03) Cao, Elliott, Joseph, Wu, Petricka, Schaefer, and Thomas Science 331,58(011)

20 Universal Shear Viscosity η (,) xt = αθ () nxt (,) High Temperature: Elliptical Flow Low Temperature: Breathing Mode

21 Breathing Mode R () t = R + A exp( t / τ ) sin( ωt + φ) 0 Damping rate 1 α 3m x = τ 0

22 Viscosity versus Energy Friction with Heating: Joins Smoothly! Cao, Elliott, Joseph, Wu, Petricka, Schaefer, and Thomas Science 331,58(011)

23 Effect of the Heating Rate w/o heating with heating

24 Effect of the Heating Rate with heating w/o heating

25 Effect of the Heating Rate Friction w/o Heating: Discontinuous!

26 Shear versus Bulk Viscosity Cao, Elliott, Wu, and Thomas New J. Phys (011)

27 Shear versus Bulk Viscosity Minimum χ 8.6 for pureα = 16.7 = B Minimum χ 1.5 for pure α = = S 4.4 Cao, Elliott, Wu, and Thomas New J. Phys (011)

28 Ratio of the Shear Viscosity to the Entropy Density η s α n αn = = s s Trap-averaged viscosity coefficient η α = s k B s/k B Average entropy per particle JLTP 150, 567 (008)

29 Entropy per Particle versus Energy per Particle Red data: Calculated nd Virial coefficient Deep trap Blue data: Measured Luo, Thomas JLTP 009 Shallow trap

30 Viscosity/Entropy 10 8 η/s η/s E/E F E/E F Cao, Elliott, Joseph, Wu, Petricka, Schaefer, and Thomas Science 331,58(011)

31 Viscosity/Entropy Cao, Elliott, Joseph, Wu, Petricka, Schaefer, and Thomas Science 331,58(011)

32 Thank You Support: ARO NSF DOE AFOSR

33 Temperature Determination The local reduced temperature is determined by the virial expansion ε = + 3 3/ nkbt (1 BnλT) B = 1/ 1/ 7/ 3/ Ho & Mueller PRL ε dx= x Utrap ( xdx ) 6 T0 σ z 5 EF σz B σfz = [1 ] 6 T σ U σ 6 σ Force balance in the trap requires: 3 3 FI Fz 0 Fz z θ 0 T T n 0 0 I = = TF TFI n0 /3