Spin Diffusion and Dynamical Defects in a Strongly interacting Fermi gas

Transcription

1 ECT Workshop, May 13 th, 2014 Spin Diffusion and Dynamical Defects in a Strongly interacting Fermi gas Tarik Yefsah Massachusetts Institute of Technology

2 10-15 m Many-body physics Length scales 1 m 10 7 m Nuclei High-T c Superfluid White Superconductors Helium dwarf Small Extremely difficult Far Can we provide a more accessible playground to benchmark theories? Can we realize clean tunable experimental systems that can directly be compared to N-Body theories?

3 Ultracold gases as practical model systems Potential Feynman s quantum simulator Interactions The diluteness of the gas is a key! n~10 14 atoms/cm 3 or 100 atoms/μm 3 Composition a million times thinner than air and more

4 When does wave mechanics matter? d n 1/3 Interparticle distance

5 When does wave mechanics matter? x d n 1/3

6 When does wave mechanics matter? x d n 1/3

7 When does wave mechanics matter? x d n 1/3

8 How cold is ultracold? Atoms move at: ~ 1 mm/s ~ 100 m/s ~ 10 6 m/s Paris in 10 seconds T(K) Ultracold atom experiments Outer space Your living room Center of the sun Supernova explosion

9 From BEC to BCS Interparticle Distance Scattering Length Weakly Interacting Bosons Strongly Interacting Bosons Strongly Interacting Fermions Weakly Interacting Fermions =

10 What can experiments teach us on BEC-BCS Crossover strongly interacting Fermi gases? Leggett (1980) Nozières & Schmitt-Rink (1985) Experiments Boulder, ENS, Innsbruck, MIT (2003) Superfluidity And Coherence + 1/k F a MIT (2005)

11 What can experiments teach us on strongly interacting Fermi gases? e.g.: ground-state energy: (at unitarity) ξ Mean-Field =0.59 Leggett Ansatz ξ advanced-theories = T-Matrix (Zwerger et al.) Epsilon expansion (Arnold, Drut & Son) Fixed-node Monte-Carlo (Giorgini et al.) ξ Experiment =0.37(1) MIT (2012) N. Navon, S. Nascimbène, F. Chevy, C. Salomon Science 328, (2010) M. Ku, A. Sommer, L. Cheuk, M. Zwierlein, Science 335, (2012)

12 Experiments as a Quantum Simulator Density equation of state 3 n f kt B Unitary Gas (Expt.) Meanfield Non- Interacting Gas M. Ku, A. Sommer, L. Cheuk, M. Zwierlein, Science 335, (2012) K. Van Houcke, F. Werner, E. Kozik, N. Prokofev, B. Svistunov, M. Ku, A. Sommer, L. Cheuk, A. Schirotzek, M. Zwierlein, Nature Physics 8, 366 (2012)

13 We know the equilibrium properties But: How about dynamics?

14 Spin Diffusion in a Strongly interacting Fermi gas

15 Large Hadron Collider (LHC)

16 4 ft

17 Little Fermi Collider (LFC) A Fermi gas collides with a cloud with resonant interactions Collision at 100 pev 23 orders of magnitude smaller than LHC Harmonic Trap A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)

18 Little Fermi Collider (LFC) A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011) Without Interactions

19 Distance [mm] Little Fermi Collider (LFC) 1.5 Without Interactions Time [ms] A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011) 60

20 Little Fermi Collider (LFC) A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011) With resonant interactions

21 mm mm 1.3 mm The bouncing gas First collision Difference density Total density Time (1ms per frame)

22 Distance [mm] Later times Time [ms] A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)

23 Distance [mm] Much later times ms Time [ms] A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)

24 Distance [mm] Much later times ms Time [ms] A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)

25 Quantum limit of spin diffusion Mean free path ~ Interparticle spacing Diffusion constant: D ~ mean free path average velocity Planck s constant D ~ = m Particle mass Quantum Limit of Diffusion In a hot relativistic fluid (e.g. Quark-Gluon Plasma): mc d md 2 T (0.1 mm) 2 = 1s D d ~ c T 2

26 Spin Diffusion vs Temperature Spin current = -D Spin density gradient dn dn js ntot d ntot sdd Ds dx dx Universal high-t behavior: Quantum Limit of Spin Diffusion 5.8(2) ħ m T T F 3/2 6.3(3) ħ m A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)

27 Dynamical defects in a Fermionic Superfluid

28 What kind of strong excitation? Dark Soliton Vortex π 2π 0 W.D Phillips group (2000) 0 π phase jump J. Dalibard group (2000) 2π phase widing Far from the ground state non-linear excitation Long-lived state stationary or topological excitation One-defect state Particle-like (energy E and effective mass M* )

29 Experiments with Fermionic Superfluids

30 Pulse off-resonant light Phase imprinting After the pulse: φ = 2Ut ħ Needs to be fast enough: t < ħ μ ~100μs superfluid Solitons in BECs by phase imprinting: Hamburg, NIST,...

31 Pulse off-resonant light Phase imprinting After time-of-flight + ramp to the BEC side of the Feshbach resonance 20μm 300μm superfluid Solitons in BECs by phase imprinting: Hamburg, NIST,...

32 Pulse off-resonant light Phase imprinting After time-of-flight + ramp to the BEC side of the Feshbach resonance 20μm 300μm piège harmonique T z = 0.1s superfluid Solitons in BECs by phase imprinting: Hamburg, NIST,...

33 Long-lived solitary wave oscillate in a superfluid Time (in s) 1s 2s 3s 4s

34 Stronger Interactions 700 G Regime of strongly interacting Molecular BEC T 4.4T 2T s z z 760 G 815 G 832 G

35 Heavy Defects BEC BdG theory for planar soliton BCS

36 An experimental riddle Long-lived solitary waves in a fermionic superfluid Inconsistent with current theories for planar solitons what did we observe?

37 In the following months Also supporting the vortex ring scenario W. Wen, C. Zhao, X. Ma, arxiv: M. D. Reichl, E. J. Mueller arxiv: A. Bulgac, M. McNeil Forbes arxiv: Against the vortex ring scenario L. Pitaevskii arxiv: From Eric Mueller s webpage

38 Origin of the debate : Integrated profile vs slice slice from numerical simulation Measured integrated density profile Z axis Bulgac et al. arxiv: Probe Probe Z axis

39 Slicing our Cloud Mask Z axis After slicing Blaster What do we see from top? Probe It is not a vortex ring

40 Tomography Top Slice ~Central slice This is not a planar but a linear defect Bottom Slice

41 Can we infer the gyroscopic nature of the defect? M. Ku, W. Ji, B. Mukherjee, E. Guardado-Sanchez, L.W. Cheuk, T. Yefsah, M.W. Zwierlein, arxiv: (2014) The trap exerts an expelling force V trap As a gyroscope the vortex experiences a Magnus force follows a trajectory along x V trap superfluid Axe z

42 Can we infer the gyroscopic nature of the defect? M. Ku, W. Ji, B. Mukherjee, E. Guardado-Sanchez, L.W. Cheuk, T. Yefsah, M.W. Zwierlein, arxiv: (2014) The trap exerts an expelling force V trap As a gyroscope the vortex experiences a Magnus force follows a trajectory along x V trap superfluid Axe z Precession as the signature of a vortex

43 Theoretical model for the vortex motion Hydrodynamic picture : v = ħ φ/2m ( φ the phase of the order parameter) Effective potential acting on the vortex E v = πħ2 4m n s r v ln( R ξ ) (valid in the limit ln( R ξ ) 1 ) (ξ vortex core size) We find Ω = 2γ+1 ω ax 8 ħω μ ln(r ξ ) At unitarity γ = 3 2 and ξ~ 1 k F For BEC Lundh, Ao, PRA 2000 Fetter, Kim, JLTP 2001

44 Heavy Defects BEC BCS

45 Heavy Defects Hydrodynamic model BEC BCS

46 Decay cascade of non linear excitations Right after imprint: Soliton? Vortex ring? 150μm 1. we observe the propagation of sound waves 2...then emerges a dark soliton that breaks into 3... a Vortex rings 4...and then the vortex ring breaks into vortices 5. Eventually, one vortex remains

47 Conclusion We observed long-lived heavy defects Excluded the vortex ring scenario It is a vortex which results from a decay cascade Benchmark for out-of-equilibrium dynamics T. Yefsah, A. Sommer, M.Ku, L. Cheuk, W. Ji, W. Bakr, M. Zwierlein, Nature 499, (2013) M. Ku, W. Ji, B. Mukherjee, E. Guardado-Sanchez, L. Cheuk, T. Yefsah, M. Zwierlein arxiv: Controlled creation of Topological excitations

48 The MIT Team Martin Zwierlein Ariel Sommer (PhD 2013) Mark Ku Waseem Bakr Lawrence Cheuk Wenjie Ji Biswaroop Mukherjee T. Yefsah, A. Sommer, M. J.-H. Ku, L. Cheuk, W. Ji, W. Bakr, M. Zwierlein, Nature 499, (2013)

49 A new kind of excitation Energy of the defect ω r ħω μ V(r) Type equation here. more 3D μ/ħω Komineas & Papanicolaou PRA 68, (2003)

50 A new kind of excitation Energy of the defect ω r ħω μ V(r) Type equation here. more 3D μ/ħω Komineas & Papanicolaou PRA 68, (2003)

51 A new kind of excitation Energy of the defect ω r ħω μ V(r) Type equation here. more 3D μ/ħω Komineas & Papanicolaou PRA 68, (2003)

52 A new kind of excitation Soliton Ground state of 2D BEC via GP equation after imprint : n mod Solitonic Vortex The S-vortex obeys a current-phase relation just like the soliton At rest the S-vortex has the far-field phase profile of a dark soliton

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55 More slides on Fermionic Superfluid

56 Speed of Sound v s mm 8.8 v s 3 F

57 Making Solitons by phase imprinting B = 815G

58 What is a soliton? Localized wave-packet Maintains its shape during propagation Dispersion (broadening) Non-linearity (localization)

59 Solitons in a Fermionic Superfluid

60 What kind of strong excitation? Dark Soliton Vortex Solitonic vortex Vortex ring W.D Phillips group (2000) J. Dalibard group (2000) Brand & Reinhardt (2001) A vortex ring maker Far from the ground state non-linear excitation Long-lived state stationary or topological excitation One-defect state Particle-like (energy E and effective mass M* )

61 What is the mass of a soliton in a Fermionic Superfluid? Wave function = pairing gap Δ(z) We do not know the current-phase relation BEC BCS (k F a) 1 GP equation still valid Bogoliubov-de Gennes equations

62 Soliton mass in the BCS limit n(x) n(z) No Cooper pairing inside soliton But fraction of Cooper pairs is very small Soliton almost completely filled with normal fluid z/ξ x/ Bare mass (= mass of missing atoms)

63 Dark Soliton in a Fermionic Superfluid (z)/ 0 z/ξ

64 Dark Soliton in a Fermionic Superfluid Andreev bound state fill the soliton (z)/ 0 z/ξ

65 Effective Mass vs Interaction Strength Solitons are filled in Effective mass M* >> M bare mass Period of oscillations should get much longer On Resonance: BEC BCS Scott, Dalfovo, Pitaevskii, Stringari, Phys. Rev. Lett. 106, (2011) Brand, Liao, PRA 83, (2011)

66 Ramp to Final Field Imaging Solitons

67 The cooling methods Laser Cooling ~ 1 mk Evaporative Cooling ~ 10 nk

68 Chu, Cohen-Tannoudji, Phillips, Pritchard, Ashkin, Lethokov, Hänsch, Schawlow, Wineland Laser Cooling Laser beams Zeeman Slower Hot atomic beam from oven, 400 C Laser cooled cloud Less than 1 mk

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70 Evaporative Cooling Invented for spin-polarized H: Hess, Kleppner, Greytak, Silvera, Walraven

71 Observation of the atom cloud Trapped Expanded Atom cloud Lens Laser beam CCD Camera Shadow image of the cloud 1 mm

72 Observation of the atom cloud Expanded Atom cloud Lens Laser beam CCD Camera Shadow image of the cloud 1 mm

73 Observation of the atom cloud Expanded Atom cloud Lens Laser beam CCD Camera Shadow image of the cloud 1 mm

74 JILA, Juni 95 (Rubidium) MIT, Sept. 95 (Sodium)

75 The cooling methods on the temperature scale Evaporative Cooling Laser to BEC cooling Room temperature Sun (center) 1 K 1 mk 1 K 100 K 10 4 K 10 6 K T Nobel Prize 2001 E. Cornell, W. Ketterle, C. Wieman Nobel Prize 1997 S. Chu, C. Cohen-Tannoudji, W. Phillips

76 Solitons as strong excitation Order Parameter Y(x) x A localized, long-lived, highly non-linear excitation An excellent probe for the medium in which it propagates

77 Heavy Solitons Aspect ratio = 6 Aspect ratio = 15 BdG theory for planar soliton BEC BCS

78 Heavy Solitons Aspect ratio = 3 Aspect ratio = 6 Aspect ratio = 15 BdG theory for planar soliton BEC BCS

79 When does wave mechanics matter? h Planck s constant Louis de Broglie Mass mv Velocity T(K) Temperature = Uncertainty of velocity 2 Size of a wave packet = Uncertainty of position Werner Heisenberg x p mt

80 How much do we need to cool? T(K) Temperature = Uncertainty of velocity 2 Size of a wave packet = Uncertainty of position Werner Heisenberg x p mt

81 Anti-Damping of Soliton Oscillations T Anti- Damping M*<0 Phase fluctuations

82 Solitons at finite temperature Period Anti-Damping Time Long period and large M*/M is a Quantum Effect Lifetime

83 Bosons versus Fermions Fermions (unsociable): Half-Integer Spin Pauli blocking Fermi sea No phase transition at low Temperature Bosons (sociable): Integer Spin Can share quantum states At low temperatures: Bose-Einstein condensation

84 Bosons N bosons sharing one and the same macroscopic matter wave (Artist s conception)

85 Fermions N fermions avoiding each other (Artist s conception)

86 Soliton effective mass phase change across soliton implies superfluid backflow Scott, Dalfovo, Pitaevskii, Stringari, PRL 106, (2011) 1 PS mn v x mn x n 2m 2 1D d 1D d 1D Effective Mass: P 1 u * S M M n1d us 2 Requires Josephson Current-Phase Relation For BEC: u s c cos 2 M s * 2M

87 Local density approximation (LDA) How to get the thermodynamics of the homogeneous gas from trapped samples? Trapped gas Homogeneous equivalent The trapping potential tunes for us the chemical potential: A single profile contains the full EoS from μ k B T = to μ 0 k B T

88 Comparison to the data T v /T axial μ/ħω

89 Comparison to the data T v /T axial T v T ax = 8 3 For BEC Lundh, Ao, PRA 2000 Fetter, Kim, JLTP 2001 μ 1 ħω ln R + 3 ξ 4 R ξ = 4 μ ħω μ/ħω

90 Comparison to the data T v /T axial T v T ax = 8 3 For BEC Lundh, Ao, PRA 2000 Fetter, Kim, JLTP 2001 μ 1 ħω ln R + 3 ξ 4 R ξ = 4 μ ħω μ/ħω T v T ax = 2 μ 1 ħω ln ξ R ξ = μ R ħω Logarithmic corrections unknown

91 Comparison to the data T v /T axial T v T ax = 8 3 For BEC Lundh, Ao, PRA 2000 Fetter, Kim, JLTP 2001 μ 1 ħω ln R + 3 ξ 4 R ξ = 4 μ ħω μ/ħω T v T ax = 2 μ 1 ħω ln ξ R ξ = R μ ħω Logarithmic corrections unknown

92 The vortex ring scenario T/T z ~1.9 Snake instability more effective than observed A. Bulgac, M. McNeil Forbes, M. M. Kelley, K. J. Roche, G. Wlazłowski Phys. Rev. Lett (2014)

93 Origin of the debate : Integrated profile vs slice slice from numerical simulation Measured integrated density profile Z axis Bulgac et al. arxiv: Probe Z axis

94 Why the data does not speak for vortex rings 1. The soliton to vortex ring decay is a stochastic mechanism But the trajectory we measure is deterministic 760 G 2. Bulgac et al. : Ts/Tz ~10 requires small vortex rings (R ~ 0.3 Rx) 815 G But the depletion we observe goes through the whole transverse size

95 Spin relaxation rate Magnetization d decays due to spin drag 2 d d sdd 0 d(t) ~ e -gt 1 g s / sd Spin relaxation gives spin drag coefficient: sd / g 10 s ~ E / F

96 Spin drag vs Temperature On resonance, spin drag must be universal: E T F sd f TF For a 50/50 mixture: Control temperature by cooling after collision 22.8 Hz 11.2 Hz 37.5 Hz