Universal quantum transport in ultracold Fermi gases

Transcription

1 Universal quantum transport in ultracold Fermi gases How slowly can spins diffuse? Tilman Enss (U Heidelberg) Rudolf Haussmann (U Konstanz) Wilhelm Zwerger (TU München) Aarhus, 26 June 24

2 Transport in quantum fluids Schäfer & Teaney 29 can mass flow without friction? F = shear viscosity to entropy ratio: expt minimum values /s ~/k B kinetic theory: s `mfp ` ~ & O() ~ k B k B s = ~ extrapolated from weak coupling gauge-gravity duality: perfect fluidity Kovtun, Son & Starinets 25 4 k B conjectured as universal lower bound transport near quantum critical point (QCP): incoherent relaxation

3 Strongly interacting Fermi gas Giorgini, Pitaevskii & Stringari 28 Bloch, Dalibard & Zwerger 28 dilute gas of and fermions, r ` Z H = dx X ~ 2 r 2 µ 2m =",# contact interaction: universal + g " # # " strong s-wave scattering, a ` (Feshbach resonance); scale invariance superfluid of fermion pairs below.2 T c /T F =.67(3) Ku et al. Science 22 Condensation T c Normal Fermi liquid Superfluid Unitarity 2 BCS /(k F a s ) BEC Attraction Randeria, Zwerger & Zwierlein 22

4 Quantum critical point resonant fixed point is Quantum Critical Point (QCP) at T=, μ=, /a= Nikolic & Sachdev 27 at unitarity /a= Enss PRA 22 abrupt change of ground state at QCP density n is order parameter: vacuum for T=, μ< gapless excitations above QCP: affect measurements in quantum critical regime

5 2 4 x (µm) 2 ρ (µm) 4 Universal properties at unitarity /a= scale invariance: properties depend only on μ/t ( angle ) Zhang+ Science 22 Figure 3 Constructing the EOS from in situ imaging. The atom cloud shown contains E F = 37 nk at the centre. a, Absorption image of the atomic cloud after quadrant ave averaging produces a low-noise density profile, n versus V. Thermometry is performed EOS (solid blue line), starting with the virial expansion for µ<.25 (green dashed the density profile yields T = 3 nk, and µ =.63. d, Given µ and T, the density profi a 3. b e.g. equation of state n = 3 T f n(µ/t ) measured by Zwierlein group (22), computed using Bold Diagrammatic MC quantum critical regime above QCP: T n /3 (T c. T. T F ) quantum and thermal fluctuations equally important, interplay challenging temperature only available scale for incoherent relaxation: Sachdev 999 n( µ,t) / n ( µ,t) van Houcke+ 22 µ /k B T Figure 4 Equation of state of the unitary Fermi gas in the normal phase. Density n ( density n and the pressure P of a non-interacting Fermi gas, versus the ratio of chem work), red filled circles: experiment (this work). The BDMC error bars are estimated u deviation systematic plus statistical errors, with the additional uncertainty from the Fe red solid lines. Black dashed line and red triangles: Theory and experiment (this work) error. Green solid line: third order virial expansion. Open squares: first order bold diag superfluid transition point from Determinental Diagrammatic Monte Carlo 3. Filled di pressure EOS (ref. 23). ~ = O() k B T ) s = 2 5 T.7 ~ (µ = ) k B NATURE PHYSICS ADVANCE ONLINE PUBLICATION 2 large-n exp n Enss 22

6 Luttinger-Ward approach repeated particle-particle scattering dominant in dilute gas: self-consistent T-matrix Haussmann 993, 994; Haussmann et al. 27 self-consistent fermion propagator (3 momenta / 3 Matsubara frequencies) spectral function A(k,ε) at Tc works above and below Tc; directly in continuum limit Tc=.6() and ξ=.36() agree with experiment Haussmann et al. 29 conserving: exactly fulfills scale invariance and Tan relations Enss PRA 22

7 Transport in linear response shear viscosity from stress correlations (cf. hydrodynamics), (!) =! Re Z ˆ xy = X p, dt e i!t Z p x p m c p c x with stress tensor (cf. Newton ) correlation function (Kubo formula): Enss, Haussmann & Zwerger, Annals Physics 2 d 3 x D ˆ xy (x,t), ˆ xy (, ) E η(ω) = (resummed to infinite order) transport via fermions and bosonic molecules: very efficient description, satisfies conservation laws, scale invariance and Tan relations Enss PRA 22 assumes no quasiparticles: beyond Boltzmann kinetic theory, works near Tc; includes pseudogap and vertex corrections

8 High-energy tails in stress correlation (shear viscosity) exact viscosity sum rule (nonperturbative check): 2 Z η(ω) [ h n] d! [ (!) tail] = P hydrodynamics... C 4 ma P +(! ) 2 C 5 p m!. ω [E F / h] T= T= 5 T= 2 T= T=.5 T=.6 Enss, Haussmann & Zwerger 2 Taylor & Randeria 2; Enss, Haussmann & Zwerger 2; Enss 23

9 η/s [ h/k B ] superfluid phonon contrib. Luttinger-Ward theory kinetic theory phonon classical limit contribution T 8 agrees with large-n transport calculation Enss PRA 22 T 3/2 lowest friction of any nonrelativistic quantum fluid T c. T/T F cf. theory: Bruun, Massignan, Schäfer, Smith,... experiment: Cao+ Science 2 Shear viscosity/entropy of the unitary Fermi gas Enss, Haussmann & Zwerger 2

10 Spin transport with ultracold gases experiment: spin-polarized clouds in harmonic trap bounce! picture: J. Thomas 2 strongly interacting gas [movie courtesy Martin Zwierlein]: A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 2 (2)

11 Spin diffusion scattering conserves total + momentum: mass current preserved but changes relative - momentum: spin current decays c Position (mm) Sommer et al. 2 spin diffusion Time (ms)

12 Spin conductivity and spin susceptibility use Einstein relation D s = s s spin conductivity: spin susceptibility: σ s me F / hn superfluid Luttinger-Ward theory classical gas kinetic theory w/o medium χ n,s / χ. superfluid χ n Sommer et al. (2) χ s Sommer et al. (2) χ n Luttinger-Ward χ s Luttinger-Ward free Fermi gas T c. T/T F T c. T/T F Enss & Haussmann PRL 22 medium effects important: large-n transport calculation Enss PRA 22 in two dimensions Enss, Küppersbusch, Fritz PRA 22 related work on spin transp.: Bruun 2; Duine et al. 2; Mink et al. 22, 23

13 Spin diffusivity obtain diffusivity from Einstein relation, D s = s s superfluid Sommer et al. (2) Luttinger-Ward theory kinetic theory (experiment rescaled from trap to infinite homogeneous box) md s / h minimum D s '.3 ~ m. T c T/T F Enss & Haussmann PRL 22 Quantum Monte Carlo simulation for finite range interaction: Wlazlowski et al. PRL 23 D s &.8 ~ m

14 Longitudinal vs transverse spin diffusion ' : J I ' ' ' ' ' ' ' ' I ' spin polarized x+dx M M - dm r b / x x+dx J : i i I i i i i : i i i I lo T(mK) Mullin & Jeon 992 M(x) M(x+dx)

15 Spin-echo experiment (Thywissen group, Toronto) M? (t) = i exp[ t 3 /24 3 ]. A (ms) M Hold time t (ms) B B-field gradient B (G/cm) Bardon+ Science 334, 722 (24) 8 D s ( /m) D: Koschorreck et al. Nature Physics Temperature (T/T F ) i

16 Spin diffusion in kinetic theory ( ) local magnetization vector and gradient M(r,t) =M(r,t) ê(r,t) Boltzmann equation for spin distribution function Dσ p Dt + i σ p t transverse i v pi M r i ê σ v pi ê r i (n p+ n p )+Ω σ p = M r i = M r i ê + M ê r i longitudinal n pσ t σ ε p ( ) σp t spin rotation coll Landau 956, Silin 957; Leggett & Rice 968-7; Lhuillier & Laloë 982; Meyerovich 985; Jeon & Mullin 988, 992 many-body T-matrix in collision integral and spin rotation Enss 23 derived as leading order in large-n expansion Enss 22

17 Spin-rotation effect diffusion equation: Leggett 97; Jeon & Mullin 988; Enss 23 M + (r,t)=m x + im y ' i (r)m + + D? ( + iµm z )r 2 M + Leggett-Rice spin-rotation effect: complex diffusion constant, spin waves; spin current precesses around effective molecular field: spin-rotation parameter µ = Ω mf τ Enss PRA 23

18 Transverse spin diffusivity (3D) r b ' : J I ' ' ' ' ' ' ' ' I ' B D s ( /m) md / h (3D) J : i i I i i i i : i i i I lo T(mK) long/trans M= trans M=. trans M=.3 trans M=.6 trans M=.9.. T/T F md / h (3D) at M= Temperature (T/T F ) i Bardon+ Science 334, 722 (24) no medium with medium medium & spin rotation... T/T F using vacuum (two-body) scattering cross section: similar to helium case medium (finite density) scattering and spin-rotation effect: diffusivity much smaller! Enss PRA 23

19 . Transverse spin diffusivity (2D) md / h (2D) at ln(k F a)= long/trans M= trans M=. trans M=.3 trans M=.6 trans M=.9.. T/T F md / h (2D) at M= ln(k F a 2D ) Koschorreck et al T=. T=., med.2 T=.5 T=.5, med ln(k F a) 2 vacuum scattering Enss PRA 23 dependence on interaction and importance of medium effects cf. η: Enss, Küppersbusch & Fritz PRA 22 trap: Chiacchiera, Davesne, Enss & Urban, PRA 23

20 Conclusion and outlook lowest friction at strong interaction: almost perfect fluidity near QCP Luttinger-Ward: Enss, Haussmann & Zwerger, Ann. Phys. 326, 77 (2) large-n: Enss, PRA 86, 366 (22) quantitative understanding of spin diffusion: Luttinger-Ward transport calculation (tail, near Tc) slowest longitudinal spin diffusivity Enss & Haussmann, PRL 9, 9533 (22) transverse spin diffusion: D can be much lower than D in degenerate, polarized gas; Leggett-Rice spin-rotation effect Enss, PRA 88, 3363 (23) on c D s &.3 ~/m outlook: spin-rotation effect (Thywissen group) 2D Fermi gas: EoS and pseudogap Bauer, Parish & Enss, PRL 2, 3532 (24) md s / h md / h (2D) at M= superfluid T c Sommer et al. (2) Luttinger-Ward theory kinetic theory. T/T F.5 T=. T=., med.2 T=.5 T=.5, med ln(k F a)

21 BKT-BCS crossover in 2D Fermi gas Bauer, Parish & Enss, PRL 2, 3532 (24) Pressure EoS (vs Turlapov data): P/P T =.2 T F T =.5 T F.4 T =.T F.3 Data from Ref. [] QMC T= ln(k F a 2D ) 4 5 Phase diagram (Tc, T*): T/TF T from Luttinger-Ward T c from Luttinger-Ward mean-field BCS.2 BCS+GMB from Ref. [2] BKT from Ref. [2]. 4 2 ln(k F a 2D ) 2 4 Tan contact density (vs Köhl data): C/k 4 F 2 Luttinger-Ward weak coupling limit Data from Ref. [3] T =QMC.5 5 x =/ ln(k F a 2D ) Spectral function/pseudogap: ρ(ω)/ρ ω/εf log (A(k, ω)) k/k F T/T F =.45 T/T F =.2 T/T F =. T/T F = ω/ε F