3D Hydrodynamics and Quasi-2D Thermodynamics in Strongly Correlated Fermi Gases. John E. Thomas NC State University

Transcription

1 3D Hydrodynamics and Quasi-D Thermodynamics in Strongly Correlated Fermi Gases John E. Thomas NC State University

2 JETLa Group J. E. Thomas Graduate Students: Ethan Elliot Willie Ong Chingyun Cheng Arun Jaganathan Nithya Arunkumar Jayampathi Kangara Lorin Baird Support: NSF ARO DOE AFOSR Research Scholars: James Joseph Ilya Arakelian

3 Outline Topics Scale Invariance in Expanding Fermi gases: Ballistic expansion of a hydrodynamic gas Shear Viscosity measurement Local viscosity from trap-averaged viscosity Spin-imalanced Quasi-two-dimensional Fermi gas Failure of true D-BCS theory D-polaron model of the thermodynamics Phase-transition to a alanced core

4 Why Study Strongly Interacting Fermi Gases? Strongly Interacting Fermionic Systems Neutron Star Quark Gluon Plasma Ultra Cold Fermi Gas Layered High Temperature Superconductors

5 Optically Trapped Fermi gas Our atom: Fermionic agnet coils

6 Feshach Resonance Singlet Diatomic Potential: Electron Spins Anti-Parallel B B 1 3 g 3 u 3 u u SCALE INVARIANT! Triplet Diatomic Potential: Electron Spins Parallel 83 G-Resonant Scattering! coll 4 db

7 Strong Interactions: Shock waves in Fermi gases Trapped gas is divided into two clouds with a repulsive optical potential. The repulsive potential is extinguished, the two clouds accelerate towards each other and collide. Shock Fronts Really strong interactions!

8 Scale Invariance in Expanding Resonant Fermi Gases coll 4 db Compressed Balloons Expanded Balloons

9 Scale-invariance: Connecting Strongly to Weakly Interacting Anti-de Sitter-Conformal Field Theory Correspondence: Connects strongly interacting fields in 4-dimensions to weakly interacting gravity in 5-dimensions. Can we connect elliptic flow of a resonant gas to the allistic flow of an ideal gas in 3D? For oth, the pressure is /3 of the energy density: p p 3 Scale Invariant? Elliptic Flow: Oserve dimensions + time

10 Measuring expanding clouds in 3D Trap with 1:3 transverse (x-y) aspect ratio Measure all three cloud radii using two cameras.

11 Transverse Aspect Ratio versus Time for a Unitary Fermi gas Energy Dependent! y 83 G 57.5 G E/E F =.5 E/E F =.75 E/E F =1. E/E F =1.69 Ballistic x

12 Scale Invariance: Ideal Gas U r r r m t Ballistic Flow v r r t Cloud average Virial Theorem: U r v m vt r r Ballistic flow t r How does the mean square radius evolve in time? z y x r Ideal gas: r (r) U trap potential

13 Defining Scale Invariant Flow r t r r x y Cloud average z m r r ru t Defines Scale Invariant Flow! t = expansion time

14 Scale Invariance: Resonant Gas Hydrodynamic gas: Elliptic flow r t r? U (r) trap potential r How does the mean square radius evolve in time? r x y z

15 Hydrodynamic Expansion From the Navier-Stokes and continuity equations, it is easy to show that a single component fluid oeys: Trap potential Pressure d dt m x i 1 3 m vi xi iu d r p S ii B N v Stream KE Shear and Bulk Viscosity Equilirium: 3 N d 3 r p r U Measured from the cloud profile and trap parameters Need to find the volume integral of the pressure: p p 3

16 Gloal Energy Conservation Just after the optical trap is aruptly extinguished*: t = + : Stream KE (t) 1 3 ε m 1 3 d r v d r ε N N Internal Energy (t) Internal Energy (t = + )

17 Scale Invariant Expansion Using the hydrodynamic equations and energy conservation it is easy to show that d dt m r r U 3 N d 3 r 3 N 3 p p d r v B Initial trap potential Conformal symmetry reaking Dp Bulk viscosity p p 3

18 Scale Invariance! Resonant gas p p 3 The ulk viscosity also vanishes so r r t m r U Ballistic Flow! Can we oserve allistic flow of an elliptically expanding gas?

19 Scale-invariant Ballistic Expansion of a Resonant Fermi gas ( t) m The aspect ratio exhiits elliptic flow, ut the mean-square cloud radius expands Ballistically: SCALE-INVARIANT! r r U r t y x Expansion time

20 Shear and Bulk Viscosity: Unitary Fermi Gas Shear viscosity Bulk viscosity n S n B B S The ulk viscosity vanishes!

21 Quantum Viscosity Viscosity: n n = density ( particles/cc) dimensionless shear viscosity coefficient Water: n = 3.3 x 1 3 n Air: n =.7 x n Fermi gas: n = 3. x n Nuclear Matter: n = 3. x 1 38? n

22 Measuring Shear Viscosity: Scaling Approximation ) ( t x x i i i ) ( v t x i i i z y x Volume scale factor ) /, /, / ( ),,, ( z y x z y x n t x y x n i i i i x / v Velocity field is linear in the spatial coordinates 3 i i ii i imag i i ii S p Q i i i x m t C t C /3 ) ( ) ( 1 3 i i x m r U Cloud-averaged shear viscosity coefficient

23 Aspect Ratio versus Expansion Time x y E/E F =.5 E/E F =.75 E/E F =1. E/E F = G Ballistic 57.5 G Shear Viscosity only Fit Parameter x y y x x y ( t) ( t)

24 Shear Viscosity at Resonance versus Reduced Temperature Transition to Superfluid

25 High Temperature Scaling of the Cloud-Averaged Viscosity 3.6 3/ Universal Scaling! Blue: New Data Red: For trap that is 5 times deeper Science 331, 58 (11)

26 Cloud Averaged Shear Viscosity versus Reduced Temperature Transition to Superfluid Reduced temperature at the trap center *EoS from Ku et al., Science, 1

27 Local Shear Viscosity: Linear Inverse Prolem For each trap averaged shear viscosity measured at reduced temperature j, we create a linear equation j i C ji i Discrete Local a i i1 Equivalent matrix equation: α Cα Can e solved iteratively α m 1 (1 β) α m β Ψ α m C Τ α Cα m

28 Determining the Local Shear Viscosity: Finite Volume Average Divergent! R C 1 3 S N d r ( r) n( r) N c 3/ 1 3/ 3 1 3/ C N Fit from our data c 1 = 3.6 Volume 4 n R 3 from theory 3/ =.77 r n r n N 3 r 3/ For a Gaussian density R C.98 r 1/

29 Image Processing Technique Iterative Matrix Inversion α m 1 (1 β) α m β Ψ α m C Τ α S Cα m

30 Iterative Solution for Test Function α m 1 (1 β) α m β Ψ α m C Τ α S Cα m q

31 Local Shear Viscosity versus Reduced Temperature High Temperature Limit =.77 θ 3/ Structure appears at low temperature

32 Local Shear Viscosity (Comparison to Theory) Measured Kinetic theory =.77 θ 3/ Enss et al., Annals 11 (Diagrammatic Kuo formula) Wlazłowski et al., PRL 1 (Monte Carlo) Guo et al., PRL 11

33 Derivative of the Local Shear Viscosity with respect to Reduced Temperature

34 Ratio of the Local Shear Viscosity to the Entropy Density* String theory limit Kovtun,Son,Starinets PRL 5 *EoS from Ku et al., Science, 1

35 Summary: 3D Hydrodynamics Scale-Invariant Strongly Interacting Fermi Gas - Tests theories of scale-invariant hydrodynamics Measured Cloud-Averaged Shear Viscosity - Single-shot method to find initial cloud size (temperature) and cloud-averaged shear viscosity self consistently Determined Local Shear Viscosity - Image processing techniques - Tests of non-perturative many-ody theory

36 Quasi-D Fermi Gases Search for high temperature superconductivity in layered materials: In copper oxide and organic films, electrons are confined in a quasi-two-dimensional geometry Complex, strongly interacting many-ody systems Phase diagrams are not well understood Exotic superfluids Enhancement of the superfluid transition temperature compared to true D materials: Heterostructures and inverse layers Quasi-D organic superconductors Intercalated structures and films of transition metals

37 Creating a Quasi-D Fermi gas Two-lowest hyperfine states of fermionic 6 Li in CO laser trap at T<. T F : 5.3 m x z Measure Column Density: n c ( x) dy nd ( x y Transverse Density Profiles: n ( ) ) D N 1 = Majority # (8 per site) N = Minority #

38 Two-Dimensional Gas r n = z n z z h z N s n = 1 n = z Two-dimensional N s d ( h ) D Density of States E h F N s True D if: E F h z D Transverse Fermi Energy

39 Quasi-Two-Dimensional Gas n = h z n = 1 N s1 1 N s n = N s True D if: 3D if: Two-dimensional E E F F h h z z Quasi-two-dimensional Quasi-D if: E What is the ground state of a quasi-d strongly-interacting Fermi gas? F h z

40 Column Densities versus N /N 1 Majority state: N 1 8 per site E = D dimer inding energy Minority state: N N 1 E F = D ideal gas Fermi energy E E F F 83 G G. 75 E E

41 Majority and Minority Radii E = D dimer inding energy E F = D ideal gas Fermi energy Ideal gas Thomas-Fermi radius - Majority E E F 6.6 E E F.75

42 Majority and Minority Radii E E F 6.6 Ideal gas E E F.75 D-BCS F E D-BCS Theory Fails! D-BCS Balanced

43 Polaron Gas Model Chevy s Fermi-polaron polaron -Collision with the 1-Fermi sea FS 1 q 1 k q k q-k pq-k, FS q kq, FS ;1 k qkfermi energy single spin down cloud of particle-hole pairs

44 D-Polaron Thermodynamics Free energy density-imalanced gas: ) ( p F F E n n n f 1 1 ) ( ) ( F m p q y E Polaron energy: Klawunn and Recati 11 ) log(1 ) ( 1 1 q q y m 1 1 n f n f Chemical potentials: f n n p 1 1 Pressure: F E q n m F Minority Polaron Energy Ideal Fermi gas

45 Polaron Energy: D-Fermi Polaron vs Analytic Approximation D Fermi-Polaron Analytic Approx. Meera M. Parish and Jesper Levinson Phys. Rev. A 87, (13) 1 ln( / F E ) Shahin Bour, Dean Lee, H.-W. Hammer, and Ulf-G. Meiner arxiv: v (14) Lianyi He, Haifeng Lu, Gaoqing Cao, Hui Hu, Xia-Ji Liu arxiv: v1 (15)

46 Majority and Minority Radii Ideal gas Polaron model D-BCS alanced E E F 6.6 E E F.75

47 Measuring the D Pressure Force alance: Harmonic Trap: P U nd ( ) U 1 ( ) m Pressure at trap center P() m N Ideal D Pressure (for density n D ) P ideal 1 n D F P() P () ideal n n ideal D () () = 1 for an Ideal Fermi gas: Determine n D () from the column densities.

48 D-Pressure Balanced Gas D-BCS p() p ideal D-polaron model

49 D-Pressure Wide Range - Polaron Model A. Turlapov, Phys. Rev. Lett. 11, 4531 (14).

50 D-Central Density Ratio E E F 6.6 E E F F E E Polaron model Ideal gas Transition to alanced core: Not predicted!

51 Summary: Quasi-D Fermi Gas D BCS theory for a True D system fails in the Quasi-D regime. D Polaron model explains several features of the density profiles in the quasi-d regime. D Polaron model with the analytic approximation is too crude to predict the transition to a alanced core. Measurements with imalanced mixtures provide the first enchmarks for predictions of the phase diagram for quasi-d Fermi gases.

52 Birthday Party Dec-14