1. PROBING HOMOGENEOUS QUANTITIES IN A TRAPPED INHOMOGENEOUS FERMI GAS FERMI SURFACE, TAN’S CONTACT AND THE SPECTRAL FUNCTION Yoav Sagi, JILA/CU, Boulder (soon, Technion,Israel) Tara Drake, Rabin Paudel, Roman Chapurin and Deborah Jin

2. The goal: Establishing a better understanding of quantum phases of interacting fermions Superfluidity, magnetic ordering, topological states, glassy phases,… The mean: ultracold Fermi gas Clean and controllable system: interactions, potential, spin composition,… Unique measurement techniques: spectroscopy, in situ imaging, momentum resolution, transport, thermodynamic, …

3. Fermionic superfluidity Fermions at two spin states: electrons, neutrons, holes, Zeeman sublevels of a fermionic isotope ( 40 K, 6 Li),… What happen when the temperature is reduced ? Weakly interacting: BCS superconductivity Below Tc: momentum space pairing around the Fermi surface. Real space pair size is very large. Pairs condense and for long range order. Above Tc: normal gapless Fermi liquid. K. Onnes discovery, 1911 T [K] Resistance

4. Strongly interacting: unconventional superconductivity Quark-Gluon plasma Neutron stars Degenerate Fermi gases High-Tc superconductors 20 orders of magnitude Universality Credit: NASA/CXC/xx;NASA/STScI;M.Weiss Credit: D. Parker, IMI, U. Birmingham Credit: Brookhaven National Laboratory Credit: D. Jin group, JILA

5. JILA’s 40 K Fermi gas machine MOT Evaporation in Cloverleaf magnetic trap Evaporation in a Crossed dipole trap The interaction energy dominates the dynamics ! Our Fano - Feshbach s-wave resonance:

6. 01 Superfluid Temperature 1/k F a BCS limitBEC limit C. A. Regal, M. Greiner, D. S. Jin, PRL. 92, 040403 (2004) M. Greiner, C. A. Regal, and D. S. Jin, Nature 426, 537 (2003) Normal Fermi liquid Molecular Bose gas

7. 01 Normal Fermi liquid PG? Superfluid Molecular Bose gas T* Temperature 1/k F a BCS limitBEC limit What is the nature of the normal state in the BCS – BEC crossover regime ? Theory Eagles, Leggett, Nozieres and Schmitt-Rink, Holland, Levin, Randeria, Strinati, Ohashi, Zwerger, Haussman, Hu, Griffin,…

8. Outline The effect of density inhomogeneity and our way to mitigate it. Observation of a sharp Fermi surface for a weakly interacting gas. Measurements of the Contact of a homogeneous unitary Fermi gas. Measurements of the occupied spectral function of a homogeneous Fermi gas in the BEC-BCS crossover regime. Is the normal state a Fermi liquid?

9. Outline The effect of density inhomogeneity and our way to mitigate it. Observation of a sharp Fermi surface for a weakly interacting gas. Measurements of the Contact of a homogeneous unitary Fermi gas. Measurements of the occupied spectral function of a homogeneous Fermi gas in the BEC-BCS crossover regime. Is the normal state a Fermi liquid?

10. Sharp features are washed out when averaging over an inhomogeneous density. Solutions: “Box” traps (Weizmann, UT at Austin, Cambridge,…), in-situ imaging (Harvard, MIT, ENS, Chicago, MPQ,…), spatial selectivity when probing. The effect of the trapping potential

11. Probing local information We optically pump the atoms in the outer parts of the cloud to a dark state. T. E. Drake, Y. Sagi, R. Paudel, J. T. Stewart, J. P. Gaebler, and D. S. Jin, PRA 86, 031601(R) (2012) hollow beam: donut beam transition m f = -9/2 -7/2 -5/2 … 4S 1/2 4P 3/2 imaging transition f = 7/2 f = 9/2  -pulse |9/2,-5/2> |11/2,-11/2> 40 K

12. Probing a homogeneous non-interacting gas The emergence of a sharp Fermi surface ! T. E. Drake, Y. Sagi, R. Paudel, J. T. Stewart, J. P. Gaebler, and D. S. Jin, PRA 86, 031601(R) (2012)

13. Outline The effect of density inhomogeneity and our way to mitigate it. Observation of a sharp Fermi surface for a weakly interacting gas. Measurements of the Contact of a homogeneous unitary Fermi gas. Measurements of the occupied spectral function of a homogeneous Fermi gas in the BEC-BCS crossover regime. Is the normal state a Fermi liquid?

14. What is the contact? S. Tan, Annals of Physics 323, 2952 (2008); Ibid., p. 2971; Ibid., p. 2987 E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008); S. Zhang and A. J. Leggett, Phys. Rev. A 79, 023601 (2009).

15. Universal relations with the contact Momentum Distribution Energy Local Pair Size Adiabatic Sweep Virial Theorem RF Lineshape S. Tan, Annals of Physics 323, 2952 (2008); Ibid., p. 2971; Ibid., p. 2987 E. Braaten and L. Platter, PRL 100, 205301 (2008); S. Zhang and A. J. Leggett, PRA 79, 023601 (2009). J. T. Stewart, J. P. Gaebler, T. E. Drake, D. S. Jin, PRL 104, 235301 (2010); E. D. Kuhnle et al. PRL 105, 070402 (2010). G. B. Partridge et al., PRL 95, 020404 (2005); F. Werner et al., EPJ B 68, 401 (2009).

16. Temperature dependence of the contact The homogeneous contact is an excellent benchmark for many-body theories ! E. D. Kuhnle et al. PRL 106, 170402 (2011)Hui Hu et al., NJP 13, 035007 (2011) Trap averageHomogeneous

17. Measuring the homogeneous contact Weakly interacting Photoemission spectroscopy (PES) m f = -9/2 -7/2 -5/2

18. Contact vs T Y. Sagi, T. E. Drake, R. Paudel, and D. S. Jin, PRL 109, 220402 (2012)

19. Contact vs T Y. Sagi, T. E. Drake, R. Paudel, and D. S. Jin, PRL 109, 220402 (2012)

20. Contact vs T Y. Sagi, T. E. Drake, R. Paudel, and D. S. Jin, PRL 109, 220402 (2012)

21. Outline The effect of density inhomogeneity and our way to mitigate it. Observation of a sharp Fermi surface for a weakly interacting gas. Measurements of the Contact of a homogeneous unitary Fermi gas. Measurements of the occupied spectral function of a homogeneous Fermi gas in the BEC-BCS crossover regime. Is the normal state a Fermi liquid?

22. Fermi liquid theory

23. Probing the many-body wavefunction m f = -9/2 -7/2 -5/2 Angle-Resolved PES (ARPES) Photoemission spectroscopy (PES) Imaging J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature 454, 744 (2008) The spectral function Fermi function

24. Photoemission Spectroscopy – limiting cases Weak Interactions Strong Interactions Molecular Limit J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature 454, 744 (2008) Molecular branch k/k F Superfluid 

25. Evidence of pseudogap with trapped 40 K J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A. Perali, P. Pieri, and G. C. Strinati, Nat. Phys. 6, 569 (2010). Hotter The true width of the dispersion might be obscured by the density inhomogeneity. Can it still be a Fermi liquid? The existence of a pseudogap phase in a strongly interacting Fermi gas remains controversial

26. Homogeneous ARPES m f = -9/2 -7/2 -5/2 Imaging

27. Homogeneous ARPES on the BEC side Purple – center of mass of the EDC, White – fit to a Gaussian There is a clear back-bending around k F

28. ARPES results around T c

29. EDCs:

30. ARPES results around T c

31. Outline The effect of density inhomogeneity and our way to mitigate it. Observation of a sharp Fermi surface for a weakly interacting gas. Measurements of the Contact of a homogeneous unitary Fermi gas. Measurements of the occupied spectral function of a homogeneous Fermi gas in the BEC-BCS crossover regime. Is the normal state a Fermi liquid?

32. Fermi liquidNon-Fermi liquid

33. Fermi liquid effective mass (BCS side) We fit the dispersion peak to a quadratic function, and extract the effective mass:

34. Summary

35. The degenerate Fermi gas team… Tara Drake, Rabin Paudel, Yoav Sagi and Roman Chapurin Deborah Jin

36. The contact and pair correlations s N 1 – number of spin up particles N 2 – number of spin down particles How many pairs are there? E. Braaten, in The BCS-BEC Crossover and the Unitary Fermi Gas, Lecture Notes in Physics, Vol. 836 (Springer, 2012). ArXiv 1008.2922. The number of pairs in a small volume is much larger than one would expect by extrapolating from larger volumes !

37. Lines: theory for homogeneous gas Symbols: averaging over the remaining density inhomogeneity

39. Theory: PRA 82, 021605(R) (2010)

40. Signature of pairing 0 1 2 01 0 1 0 1 E/E F k/k F Non-interacting gasNormal Fermi liquid BCS superfluid kFkF kFkF 

41. Does a Fermi gas has PG phase ? Experiments: Thermodynamics : not a sensitive probe - ? Transport: Duke experiment measures low viscosity -> no well defined quasi-particles. - YES RF spectroscopy (JILA): evidence of pairing in the normal state. -YES P. Magierski, G. Wlazłowski, A. Bulgac, PRL 107, 145304 (2011). Theories: most predict a pseudogap at unitarity. G 0 G 0, GG 0, Virial, QMC – YES GG - NO

42. Width dependence on momentum Near the phase transition, at different interaction strength On the BEC side, at different temperatures In these figures we plot the full width at half the maximum:

43. Comparison with Fermi liquid theory – averaging over the remaining inhomogeneity BCS Unitarity BEC vv Looking around the Fermi surface v v

44. Homogeneous condensate fraction at unitarity

45. High-Tc superconductors versus strongly interacting Fermi gases Credit: Laboratoire National des Champs Magnétiques Intenses, Toulouse, France Credit: HIGH ENERGY ACCELERATOR RESEARCH ORGANIZATION, KEK

46. Controlling the interaction Magnetic scattering resonance (Fano-Feshbach) New molecular bound state leads to a divergence of the scattering properties!

47. Strong interactions When is the gas strongly interacting? Generally, there is no small parameter and the system cannot be described by mean field theories. The interaction energy dominates the dynamics !

48. Fermionic condensation M. Greiner, C. A. Regal, and D. S. Jin, Nature 426, 537 (2003) C. A. Regal, M. Greiner, D. S. Jin, PRL. 92, 040403 (2004)

49. Probing a homogeneous gas We fit to a homogeneous Fermi-Dirac distribution:

50. The probability to scatter a photon We model the optical pumping with a two-level open system: - Rabi frequency - Excited state lifetime - Branching ratio We solve using the optical Bloch equations:

51. Hollow beam propagation Assumption: each scattering event results in the removal of one photon and one atom: The probability to scatter a photonNumber of atoms The change in the number of photons: Atoms Hollow beam

52. Angle Resolved Photo-Emission Spectroscopy (ARPES) Raw Signal Conservation of energy and momentum Measures the occupied part of the single-particle spectral function in the energy-momentum space.

53. Crossover theories I … T~T c

54. Crossover theories II

55. Crossover theories III NSRBCS- Leggett NSRBCS- Leggett

56. Crossover theories IV T/T F =0.0 1 0.060.14 0.16 (T c )0.180.3 Luttinger-Ward formalism

57. Other experiments - thermodynamics S. Nascimbene et al. (ENS), Nature 463, 1057 (2010) M. J. H. Ku et al. (MIT), arXiv: 1110.3309 (2011) Also, spin transport measurements are not conclusive (Sommer et al. Nature 472, 201, 2011).

58. A tale of two tails… J. T. Stewart, J. P. Gaebler, T. E. Drake, D. S. Jin, PRL 104, 235301 (2010) F. Werner, L. Tarruell, Y. Castin, Euro. Phys. J. B 68, 401 (2009)

59. Universal energy relations J. T. Stewart, J. P. Gaebler, T. E. Drake, D. S. Jin, PRL 104, 235301 (2010)

60. Contact vs. fraction Symbols=full local density calculation

61. Pairing pseudogap in high-T c SC Suppression of low-energy spectral weight due to incoherent pairing in the normal state. Tunneling Spectra (DOS) of underdoped Bi 2 Sr 2 CaCu 2 O 8. Renner et al., PRL 80, 149 (1998). TcTc DOS Energy Momentum ARPES spectra of Bi 2 Sr 2 CaCu 2 O 8 at 140K>T c =90K. Kanigel et al., PRL 101, 137002 (2008).