Quantum limited spin transport in ultracold atomic gases

Transcription

1 Quantum limited spin transport in ultracold atomic gases Searching for the perfect SPIN fluid... Tilman Enss (Uni Heidelberg) Rudolf Haussmann (Uni Konstanz) Wilhelm Zwerger (TU München) Technical University Darmstadt, 7 February 2013

2 Is an ideal fluid realized in Nature? [Schäfer, Teaney 2009] flow without friction? vanishing shear viscosity η? some pictures are removed for copyright reasons--sorry F = kinetic theory (Boltzmann equation) for dilute gas: η measures momentum transport = 1 3 n p `mfp, `mfp = 1 n : ' p mkb T (T ) grows with T

3 How about a superfluid? [Schäfer, Teaney 2009] superfluid helium-4: SF =0 phonon contribution [Heikkilä, Hollis-Hallett 1955] [Landau, Khalatnikov 1949] generically, η has minimum at strong coupling: universal bounds on transport coefficients? other sources of dissipation vanish for certain fluids (e.g., bulk viscosity ζ=0 in scale-invariant fluids) but η is always nonzero

4 Estimating the shear viscosity shear viscosity η on vastly different scales: normalize by entropy density s, s =# ~ k B 1 (ħ indicates quantum effect) degenerate quantum gas: Fermi momentum p ' ~k F ' ~/` cross section limited by unitarity 3 np`mfp, s ' k B n =) s ' `mfp ` apple 4 k 2 ' `2 ~ k B mean free path `mfp =1/(n ) & ` (in absence of localization) =) s & ~ k B (beyond kinetic theory: strong coupling)

5 Insights from string theory holographic duality: conformal field theory (CFT) dual to AdS 5 black hole: shear viscosity CFT entropy graviton absorption cross section (~ area of event horizon) Hawking-Bekenstein entropy (~ area of event horizon) N =4 specifically SU(N), SYM theory (no confinement, no running coupling) in strong-coupling t Hooft limit is dual to classical gravity: = g 2 N s ~ 4 k B [Policastro, Son, Starinets 2001; Kovtun, Son, Starinets 2005] conjecture of universal lower bound: perfect fluidity

6 Unitary Fermi gas two-component Fermi gas, with contact interaction Z X S = d d xd h r 2 i 2m µ + g " # # " scattering amplitude (3d) 1 f(k) = 1/a ik + r e k 2 /2 strong scattering in unitary limit 1/a =0: f(k! 0) = i k universal for dilute system (broad resonance) r e n 1/3 =",# superfluid of fermion pairs below [Sa de Melo, Physics Today 2008] T c /T F 0.16 [Ku et al. Science 2012] [Ketterle 2005]

7 Luttinger-Ward theory Luttinger-Ward (2PI) computation: repeated particle-particle scattering self-consistent T-matrix self-consistent fermion propagator (300 momenta / 300 Matsubara frequencies) spectral fct. A(k,ε) at Tc equation of state: pressure experiment: Tc=0.167(13), ξ=0.370(5)(8) [Ku et al. 2012, Zürn et al. 2012] [Haussmann et al. 2009] [Haussmann et al. 2007] Luttinger-Ward: Tc=0.16(1), ξ=0.36(1)

8 Viscosity in linear response: Kubo formula viscosity from stress correlations (cf. hydrodynamics): (!) = 1! Re Z 1 ˆ xy = X p, p x p m c p c x with stress tensor (cf. Newton ) correlation function (Kubo formula): [Enss, Haussmann, Zwerger Ann. Phys. 2011] η(ω) = 0 dt e i!t Z transport via fermions and bosonic molecules: very efficient description, satisfies conservation laws (exact scale invariance and Tan relations [Enss 2012]) d 3 x D ˆ xy (x,t), ˆ xy (0, 0) E (resummed to infinite order) assumes no quasiparticles: beyond Boltzmann

9 ( ) [ h n] hydrodynamics P 1+(! ) 2 T=10 T= 5 T= 2 T= 1 T=0.5 T=0.151 C p m! [E F / h] Viscosity spectral function [Enss, Haussmann, Zwerger 2011]

10 Contact coefficient generically, short-distance (UV) behavior depends on non-universal details of interaction potential r 0 k 1 for zero-range interaction ( at most two particles within distance F r0 ) this becomes universal:, all others far away (medium) two-particle density matrix for : Z d 3 R r 0 <r k 1 D " (R + r 2 ) # (R r 2 ) # (R r 2 ) " (R + r 2 ) E = C F many-body 1 r few-body 2 1 a Tan contact C: probability of finding up and down close together (property of strongly coupled medium) [Tan 2005]

11 Contact coefficient determine C: lim p!1 n p = C p 4 [Stewart et al. 2010] intuitively: absorb external perturbation with large energy/momentum far away from coherent peak of a single particle need to hit 2 particles close together to give energy+momentum to both absorption rate ~C access strong coupling at arbitrary temperature via perturbation theory, predictive power (cf. Landau parameters)

12 Viscosity tail analytical high-frequency tail [Enss, Haussmann, Zwerger 2011] (!!1)= viscosity sum rule 2 Z 1 0 ~3/2 C 15 p m! d! [ (!) tail] = P Contact coefficient C [k F 4 ] ~ 2 C 4 ma Tan contact C visc. tail 15 C T [T F ] provides non-perturbative check [Enss, Haussmann, Zwerger 2011; cf. Taylor, Randeria 2010]

13 High-temperature limit ( ) [ h n] hydrodynamics T=10 T= 5 T= 2 T= 1 T=0.5 T=0.151 high temperature T TF (virial expansion): η(ω) = [E F / h] (! = 0) = 45 3/2 64 p 2 ~n T T F 3/2 vertex corrections crucial agrees exactly with Boltzmann result [Massignan et al. 2005]

14 /s [ h / k B ] phonon classical full Kubo phonon contribution T 8 T 3/2 classical limit 0.5 Tc T [T F ] Shear viscosity/entropy of the unitary Fermi gas [Enss, Haussmann, Zwerger 2011]

15 Shear viscosity bounds bound from stochastic hydrodynamics: [Romatschke, Young arxiv: ] kinetic theory η/n phonons disallowed region η bound incl. MIT EoS Enss et al T/T F [see also Schäfer; Bruun, Smith PRA 2007 (kin), Enss PRA 2012 (large-n), Wlazlowski et al. PRL 2012 (QMC), Kryjevski arxiv: (ε expansion), Schäfer, Chafin arxiv: (hydro)]

16 How about spin transport? experiment: spin-polarized clouds in harmonic trap bounce! [pic J. Thomas 2011] strongly interacting gas [movie courtesy Martin Zwierlein]: [A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)]

17 Is there a quantum bound for spin diffusion? scattering conserves total + momentum: mass current preserved but changes relative - momentum: spin current decays c Position (mm) kinetic theory: diffusion coefficient [Sommer et al.; Bruun NJP 2011] Fermi velocity Time (ms) spin diffusion D s v`mfp v ' ~k F m `mfp = 1 n ' 1 ' 1 k F mean free path with cross section (unitarity) k 2 F =) D s ' ~ m quantum limit for diffusion

18 Spin diffusivity cold atom experiment: D s = area time (100 µm)2 (1 second) ~ m 100 md s / D s & 6.3 ~ m T/T F [Sommer et al. 2011] solid state: spin Coulomb drag in GaAs quantum wells D s ' 500 ~ m [Weber 2005]

19 Computing the spin diffusivity Luttinger-Ward (2PI) theory: use Einstein relation D s = s s spin conductivity s(q,!) from current correlation fct. h[j " j #, j " j # ]i include vertex pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi corrections to satisfy, particle number conservation 10 ( ) η/ hn 1 classical gas w/ Pauli blocking Pauli blocking + medium scattering T c 10 T/T F importance of medium effects (2d): [Enss, Küppersbusch, Fritz PRA 2012] md/ h 10 1 classical gas w/ Pauli blocking Pauli blocking + medium scattering T c 10 T/T F

20 Dynamical spin conductivity σ s (ω) me F / hn ! 3/ Luttinger-Ward theory Drude model Γ/(ω 2 +Γ 2 ) universal tail Cω -3/2 1 hω/e F exact high-frequency tail [Hofmann PRA 2011; Enss, Haussmann PRL 2012] s(!!1)= C 3 (m!) 3/2 satisfies spin sum rule despite tail [Enss, EPJ Spec.Topics 2013] Z d! s(!) = n m

21 Spin conductivity and susceptibility χ n,s / χ superfluid χ n Sommer et al. (2011) χ s Sommer et al. (2011) χ n Luttinger-Ward χ s Luttinger-Ward free Fermi gas T c T/T 10 F [Enss, Haussmann PRL 2012]

22 Spin diffusivity obtain diffusivity from conductivity, D s = s (! = 0) s 10 superfluid Sommer et al. (2011) Luttinger-Ward theory kinetic theory (experiment rescaled from trap to infinite homogeneous box) md s / h minimum D s ' 1.3 ~ m 1 T c T/T 10 F [Enss, Haussmann PRL 2012] recent Monte Carlo simulation for finite system: [Wlazlowski et al. arxiv: ] D s & 0.8 ~ m

23 Conclusion and outlook universal viscosity bound: unitary Fermi gas most perfect non-relativistic fluid transport calculation beyond Boltzmann (tail, no qp) η/s [ h / k B ] phonon classical full Kubo clouds of opposite spin bounce off each other: T [T F ] quantitative understanding of spin diffusion: unitary spin diffusivity D s & 1.3 ~/m bound from holographic duality? md s / h 10 superfluid Sommer et al. (2011) Luttinger-Ward theory kinetic theory 1 challenges: modeling of trap, local transport measurements extract diffusivity from spin-resolved dynamic structure factor T c T/T 10