Yoav Sagi , JILA/CU, Boulder (soon , Technion ,Israel)

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 (40K, 6Li),… What happen when the temperature is reduced ? Weakly interacting: BCS superconductivity K. Onnes discovery, 1911 • 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. Resistance T [K]

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

5. JILA’s 40K 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. 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 Temperature -1 0 1 Superfluid BCS limit BEC limit 1/kFa

7. 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,… Normal Fermi liquid Molecular Bose gas T* PG? Temperature -1 0 1 Superfluid BCS limit BEC limit 1/kFa

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. The effect of the trapping potential • 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.

11. Probing local information • We optically pump the atoms in the outer parts of the cloud to a dark state. 40K |9/2,-5/2> |11/2,-11/2> 4P3/2 donut beam transition imaging transition p-pulse f = 7/2 f = 9/2 hollow beam: 4S1/2 mf = -9/2 -7/2 -5/2 … T. E. Drake, Y. Sagi, R. Paudel, J. T. Stewart, J. P. Gaebler, and D. S. Jin, PRA 86, 031601(R) (2012)

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. Partridgeet 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 ! Trap average Homogeneous E. D. Kuhnle et al. PRL 106, 170402 (2011) Hui Hu et al., NJP 13, 035007 (2011)

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

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 Angle-Resolved PES (ARPES) Photoemission spectroscopy (PES) mf = -9/2 -7/2 -5/2 Fermi function The spectral function Imaging J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature 454, 744 (2008)

24. Photoemission Spectroscopy – limiting cases Weak Interactions Molecular Limit Superfluid Strong Interactions Molecular branch 2D k/kF J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature 454, 744 (2008)

25. Evidence of pseudogap with trapped 40K 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 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).

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

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 kF

28. ARPES results around Tc

29. ARPES results around Tc EDCs:

30. ARPES results around Tc

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. Is the normal state a Fermi liquid? Fermi liquid Non-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… Deborah Jin Tara Drake, Rabin Paudel , Yoav Sagi and Roman Chapurin Thank you for your attention!

36. The contact and pair correlations N1 – number of spin up particles N2 – number of spin down particles How many pairs are there? s The number of pairs in a small volume is much larger than one would expect by extrapolating from larger volumes ! E. Braaten, in The BCS-BEC Crossover and the Unitary Fermi Gas, Lecture Notes in Physics, Vol. 836 (Springer, 2012). ArXiv 1008.2922.

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

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

40. Signature of pairing Non-interacting gas Normal Fermi liquid BCS superfluid kF kF 2 2D 1 E/EF 0 1 0 1 1 0 0 k/kF k/kF k/kF

41. Does a Fermi gas has PG phase ? Theories: most predict a pseudogap at unitarity. G0G0, GG0, Virial, QMC – YES GG - NO 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).

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

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