1. C. Salomon Saclay, June 2, 2010 Thermodynamics of a Tunable Fermi Gas

2. The ENS Fermi Gas Team S. Nascimbène, N. Navon, K. Jiang, F. Chevy, C. S. L. Tarruell, M. Teichmann, J. McKeever, K. Magalhães, A.Ridinger, T. Salez, S. Chaudhuri, U. Eismann, D. Wilkowski, F. Chevy, Y. Castin, M. Antezza, C. Salomon Theory collaborators: D. Petrov, G. Shlyapnikov, R. Papoular, J. Dalibard, R. Combescot, C. Mora C. Lobo, S. Stringari, I. Carusotto, L. Dao, O. Parcollet, C. Kollath, J.S. Bernier, L. De Leo, M. Köhl, A. Georges

3. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA The Equation of State of a Fermi Gas with Tunable Interactions Cold atoms, Spin ½ Dilute gas : 10 14 at/cm 3, T=100nK BEC-BCS crossover Spin imbalance, exotic phases Neutron star, Spin ½ G. Baym, J. Carlson, G. Bertsch,…

4. E B =-h 2 /ma 2 Lithium 6 Feshbach resonance

5. BCS phase condensate of molecules BEC-BCS Crossover No bound state Bound state Tuning interactions in Fermi gases Lithium 6 Tuning interactions in Fermi gases Lithium 6 a>0 a<0

6. Experimental sequence - Loading of 6 Li in the optical trap - Tune magnetic field to Feshbach resonance - Evaporation of 6 Li - Image of 6 Li in-situ

7. Spin balanced Unitary Fermi Gas Pixels Optical Density (a.u)

8. Direct proof of superfluidity MIT 2006 Critical SF temperature = 0.19 T F

9. Thermodynamics of a Fermi gas Variables :scattering lengtha temperatureT chemical potential µ We build the dimensionless parameters : We have measured the EoS of the homogeneous Fermi gas Interaction parameter Fugacity (inverse) Canonical analogs Pressure contains all the thermodynamic information

10. Local density approximation: gas locally homogeneous at Measuring the EoS of the Homogeneous Gas i =1, spin up i =2, spin down

11. Extraction of the pressure from in situ images doubly-integrated density profiles equation of state measured for all values of Ho, T.L. & Zhou, Q., Nature Physics, 09 Measuring the EoS of the Homogeneous Gas

12. The Equation of State at unitarity Thermodynamics is universal J. Ho, E. Mueller, ‘04 S. Nascimbene et al., Nature, 463, 1057, (Feb. 2010)

13. Phase diagram: exploring two fundamental sectors Phase diagram: exploring two fundamental sectors : balanced gas at finite T,  1 =  2 =  : imbalanced gas at T=0 Inverse of the fugacity

14. Equation of state of balanced gas Accuracy: 6% High temperature Low temperature Superfluid region

15. High T : virial expansion X. Liu et al., PRL 102, 160401 (2009) G. Rupak, PRL 98, 90403 (2007) No theoretical prediction  4-body problem SF

16. Comparison with Many-Body Theories (1) E. Burovski et. al., PRL 96, 160402 (2006) Diagram. MC A. Bulgac et al., PRL 99, 120401 (2006) QMC R. Haussmann. et al., PRA 75, 023610 (2007) Diagram.+analytic

17. Comparison with Many-Body Theories (2) E. Burovski et. al., PRL 96, 160402 (2006) Diagram. MC A. Bulgac et al., PRL 99, 120401 (2006) QMC R. Haussmann. et al., PRA 75, 023610 (2007) Diagram.+analytic New Q. MC by Amherst in good agreement for R. Combescot, Alzetta, Leyronas, PRA, 09

18. Low Temperature A. Bulgac et al., PRL 99, 120401 (2006) R. Haussmann. et al., PRA 75, 023610 (2007) Normal phase : Landau theory of the Fermi liquid ? Superfluid at T = 0 B. Svistunov, Prokofiev, 2006 we find : C. Lobo et al., PRL 97, 200403 (2006) Normal phase Exp. data

19. Normal-Superfluid phase transition We find the critical parameters E. Burovski et. al., PRL 96, 160402 (2006) R. Haussmann. et al., PRA 75, 023610 (2007) K.B. Gubbels and H.T.C Stoof, PRL 100, 140407 (2008) A. Bulgac et al., PRL 99, 120401 (2006) Fermi liquid of quasiparticles Superfluid transition also Good agreement with theory, with Riedl et al., and with M. Horikoshi, et al. Science 327, 442 (2010);  /E F  c = 0.49 (2)

20. What happens to superfluidity with imbalanced Fermi Spheres ? Superconductors: apply an external magnetic field but Meissner effect Cold Atoms: change spin populations Definition :     P: spin polarization A question discussed extensively since the BCS theory and more than 30 papers in the last 3 years MIT ’06,: 3 phases, RICE ’06: 2 phases, ENS ’09: 3 phases

21. Exploring the spin imbalanced gas at zero temperature Exploring the spin imbalanced gas at zero temperature : balanced gas at finite T : imbalanced gas at T=0 Inverse of the fugacity

22. Equation of state h( , 0) i.e.(T=0) Deviation from h s at T=0 SF-Normal Phase Transition MIT: Y. Shin, PRA 08 Fixed-Node SF Mixed normal phase Ideal gas

23. An interesting limit: the Fermi Polaron Partially polarized normal phase - Easier to understand in the limiting case of a single minority atom immersed in a majority Fermi sea : the Fermi polaron - Proposed by Trento, Amherst, Paris - Observed at MIT by RF spectroscopy - Can we describe the normal phase as a Fermi liquid of polarons ? binding energy of a polaron in the Fermi sea effective mass Schirotzek et. al, PRL 101 (2009)

24. Ideal gas of polarons EOS for a Fermi liquid of polarons Using : ENSMIT Pilati & Giorgini Lobo et. al. Combescot & Giraud Fixed Node (Lobo et. al.) Mixed normal phase: Ideal gas + ideal gas of polarons

25. The Equation of State in the BEC-BCS crossover The ground state: T=0 N. Navon, S. Nascimbene, F. Chevy and C. Salomon, Science 328, 729 (2010)

26. Single-component Fermi gas: Two-component Fermi gas δ 1 : grand-canonical analog of η: chemical potential imbalance Ground state of a tunable Fermi gas

27. Equations of State in the Crossover First –order phase transitions: slope of h is discontinuous Paired SF polarized Normal phase Ideal F Gas Paired SF

28. Phase diagram BCS BEC

29. SF We use the most advanced calculations for Prokof’ev et al., R. Combescot et al, BEC side, unitarity: excellent agreement BCS side: deviation close to η c Comparison with the two ideal gases model

30. Full pairing: Symmetric parametrization: Superfluid Equation of State

31. Superfluid Equation of State in the Crossover BCS BEC 1/a= 0

32. BCS limit: BEC limit Unitary limit Lee-Yang correction mean-field molecular binding energy mean-field Lee-Huang-Yang correction We get:  s = 0.41(1) contact coefficient  = 0.93(5) Asymptotic behaviors

33. Fit of the LHY coefficient: 4.4(5) theory: No effect of the composite nature of the dimers X. Leyronas et al, PRL 99, 170402 (2007) mean-field LHY Measurement of the Lee-Huang-Yang correction BCS BEC

34. For elementary bosons: B is not universal for elementary bosons (Efimov physics) Λ * : three-body parameter E. Braaten et al, PRL 88, 040401 (2002) Here: universal value (using an appropriate Padé fit function) Beyond the Lee-Huang-Yang correction

35. Direct Comparison to Many-Body Theories Grand-Canonical – Canonical Ensemble Direct Comparison to Many-Body Theories Grand-Canonical – Canonical Ensemble Chang et al, PRA 70, 43602 (2004) Astrakharchik et al, PRL 93, 200404 (2004) Pilati et al, PRL 100, 030401 (2008) Fixed-Node Monte-Carlo theories Nozières-Schmitt-Rink approximation Hu et al, EPL 74, 574 (2006) Diagrammatic theory Haussmann et al, PRA 75, 23610 (2007) Quantum Monte Carlo Bulgac et al, PRA 78, 23625 (2008) Exp.

36. Conclusion - Perspectives - EOS of a uniform Fermi gas at unitarity in two sectors 1) balanced gas at finite T 2) T = 0 imbalanced gas - Precision Test of Many-body Theories - Next: Mapping the EOS in the complete space  imbalanced gas at finite T, mass imbalance - Lattice experiments - EoS in the BEC-BCS crossover at T=0 - First quantitative measurement of Lee- Huang-Yang quantum corrections and Lee-Yang on BCS side - Simple description of the normal phase as two ideal gases on BEC and unitary; breakdown on BCS side

38. Thermodynamics of a unitary Fermi gas - Equation of state of a homogeneous two-component Fermi gas (appropriate variables experimentally) 2) Trapping potential  inhomogeneous gas Measured values averaged over the trap 1) Thermometry of a strongly correlated system : difficult ! Two problems  immersing weakly interacting bosonic 7 Li (« ideal thermometer »)

39. Conclusion Measurement of the EoS : - Unitary gas at finite temperature - Fermi gas T=0 in the BEC-BCS crossover Quantitative many-body physics – benchmark for theories

40. Comparison with Tokyo group Grand-canonical Canonical ensemble Viriel 2 ? Disagrees with Viriel 2 expansion M. Horikoshi, et al. Science 327, 442 (2010); Tokyo ENS

41. Typical images One experimental run : density profile + temperature - Image of 6 Li in-situ Image of 7 Li in TOF

42. Conclusion Measurement of the EoS : - Unitary gas at finite temperature - Fermi gas T=0 in the BEC-BCS crossover Quantitative many-body physics – benchmark for theories

43. Trapped gas : Results We find : Boulder Duke Tokyo Innsbruck

44. Perspectives – Open Questions - Critical temperature in the BEC-BCS crossover - Nature of the Normal phase in the crossover - Low-lying excitations of the superfluid Fermi liquid Pseudo-gap Superfluid

45. Neutron characteristics spin ½ scattering length effective range Universal regime: « dilute » limit (mean density ) Tc=10 10 K =1 MeV, T=T F /100 k F a ~ -4,-10,… Simulation of Neutron Matter

46. k/k F 1 n(k) 1 k/k F 1 n(k) 1 Thermodynamics Is a useful but incomplete equation of state ! Complete information is given by thermodynamic potentials: Equ. of state useful for engines, chemistry, phase transitions,…. Grand potential Entropy PressureTemperature Atom number Chemical potential Volume Internal energy We have measured the grand potential of a tunable Fermi gas S. Nascimbene et al., Nature, 463, 1057, (Feb. 2010), arxiv 0911.0747 N. Navon et al., Science 328, 729 (2010)

47. MIT and Rice experiments Obvious phase separation seen on the in situ optical density difference Clogston limit at MIT is P= 0.75 and close to 1 at Rice MIT: M. Zwierlein, A. Schirotzek, C. Schunck, and W. Ketterle, Science 311, 492 (2006). RICE: G. Partridge, W. Li, R. Kamar, Y. Liao, and R. Hulet, Science 311, 503 (2006). Difference of optical density of the two spin populations

48. Direct comparison to many-body theories Grand-Canonical – Canonical Ensemble Direct comparison to many-body theories Grand-Canonical – Canonical Ensemble Chang et al, PRA 70, 43602 (2004) Astrakharchik et al, PRL 93, 200404 (2004) Pilati et al, PRL 100, 030401 (2008) Fixed-Node Monte-Carlo theories Nozières-Schmitt-Rink approximation Hu et al, EPL 74, 574 (2006) Diagrammatic theory Haussmann et al, PRA 75, 23610 (2007) Quantum Monte Carlo Bulgac et al, PRA 78, 23625 (2008)

49. We fit our data in the region Simple analytical theory R. Combescot et al, PRL 98, 180402 (2007) Fixed Node Monte Carlo S. Pilati et al, PRL 100, 030401 (2008) Diagrammatic Monte Carlo N. Prokof’ev et al, PRB 77, 020408 (2008) Most advanced analytical theory R. Combescot et al, EPL 88, 60007 (2009) Polaron effective mass Collective modes S.Nascimbene. et al, PRL 103, 170304 (2009) MIT measurement Y. Shin, PRA 77, 041603 (2008)

50. Determination of  0 Density profile  segment 7 Li Unknown ! No model-independent determination of  0 ! Progressive construction of P by connecting adjacent segments From high temperature side assuming viriel 2 coefficient of Free parameter for each image  0  Y axis value fixed, X axis rescaled by a free factor

51. Construction of the EOS

52. superfluid core normal phase Superfluid core for We calculate using fits of our EoS Application: Critical polarization for superfluidity EOS MIT

53. Partially polarized phase Simple physical picture: one builds a Fermi sea of fermionic polarons Single-polaron ground state: Excited states: Then Fermi pressure of majority atoms Fermi pressure of polarons

54. Using 7 Li as a thermometer 6 Li 7 Li - Feshbach resonance @ 834 G a 76 = 2 nm  Good ! Weakly interacting a 77 = -3 nm  Not so good… BEC instability  Low temperature limited by collapse : - With typical atom numbers close to expected T c for SF transition…

55. Clogston-Chandrasekhar limit - Naive argument using BCS picture : the energy of excess particles must be compared with « robustness » of the fermion pairs : : SF is stable with equal densities ? - Relation predicted by BCS theory :

56. Density Profiles - Data consistent with 3 phases + LDA as MIT - In agreement with theory (3 phases) -- Solid line: model with approximate eq of state ( Recati et al., 2008) - In-situ imaging at high field of 2 spin states Vertical width = 40 µm SF core Partially polarized normal phase Fully polarized shell

57. Previous works: superfluid equation of state Radial breathing mode experiments Altmeyer et al, PRL 98, 040401 (2006) theory Mean field QMC calculations + hydrodynamics + scaling ansatz Astrakharchik et al, PRL 95, 030404 (2005)

58. Extracting the BCS asymptotic behavior On the BCS side: Pade approximant mean-field value used as a constraint From : Lee-Yang coefficient: 0.18(2) theory: mean-field mean-field + LY

59. Using and, one calculates at : jump of n 2 /n 1 low imbalance phase: n 2 =n 1 large imbalance phase: 0<n 2 <n 1 Calculation of the density ratio

60. Thermodynamics of the unitary gas From : agrees with previous measurements Related to short-range pair correlations Two weeks ago (Hu et al, arXiv:1001.3200): Dynamic structure factor measurement

61. -- One example of a cold atom system at the interface with condensed matter: surprising richness of the simplest of strongly correlated systems. Precision measurement of the EOS. Useful to understand other more complex quantum many-body systems. -- In progress: General equation of state at finite temperature (ongoing !) Superfluid temperature = f(k F a), polaron oscillation (PRL’ 09) -- FFLO phase,… -- Lattice experiments and low D systems Prospects

62. A good reading ! Enrico Fermi on lake Como

64. Experimental approach Experimental approach Cooling of 7Li and 6Li 1000 K: oven 1 mK: laser cooling 10  K: evaporative cooling in magnetic trap E= - .B Ioffe-Pritchard trap

65. New experimental methods: Image a many-body wavefunction with micrometer resolution in optical lattice Measure correlation functions Photoemission spectroscopy to measure Fermi surface and single particle excitations, Dao et al., 07 Cooling in optical lattices: J.S. Bernier et al. 09, J. Ho 09,.. Time-dependent phenomena in 1, 2, and 3 D Other many-body Hamiltonians Bosons, fermions, and mixtures, 6 Li— 40 K Periodic potential or disordered (Anderson loc.) Gauge field with rotation or geometrical phase Quantum Hall physics and Laughlin states Non abelian Gauge field for simulating the Hamiltonian of strong interactions in particle physics Quantum simulation with cold atoms Period: 4.9  m D. Weiss et al., Nature Phys, 2007

66. Dispersion relation of polarons Low energy spectrum of the polaron: Within LDA: local Fermi energy of majority species: The quasi-particle evolves in an effective potential: Oscillation frequency of the polaron: Measuring the oscillation frequency of polarons gives access to the effective mass m* A: binding energy of a polaron= - 0.60 F. Chevy `06, Lobo et al., Prokofiev, Svistunov, MC N. Nascimbene, N. Navon, K. Jiang, L. Tarruell, J. Mc Keever, M. Teichmann, F. Chevy, C. Salomon, to appear in PRL 2009 arXiv:0907.3032

67. ExperimentENS ( 6 Li)0.42(15) Rice ( 6 Li)0.46(5) JILA( 40 K)0.46(10) Innsbruck ( 6 Li)0.27(10) Duke ( 6 Li)0.51(4) TheoryBCS0.59 Astrakharchik0.42(1) Perali0.455 Carlson0.42(1) Haussmann0.36 Determination of  Universal equation of state of the unitary Fermi gas at zero temperature For  is a universal number

68. Double-integrated profiles of density difference MIT Flat top = hollow core with LDA = paired core RICE Non monotonic behaviour : LDA violating feature Effect of the number of atoms, aspect ratio and temperature ?

69. Local chemical potentials In-situ density profiles - easy to interpret at T=0 - Trapping potential is useful (this time !) - under LDA, the trapping provides us with locally homogeneous gases with different values of : in one image, we have the full T=0 phase diagram of the system ! Superfluid Normal partially polarized Fully polarized Majority Difference Minority Column density Theory, F. Chevy ‘06, Lobo et al.,

70. Typical images One experimental run : we get density profile + temperature Density profile of 6 LiTemperature 6 Li 7 Li Mixture remarkably stable !

71. Determination of  0 Density profile  segment 7 Li Unknown ! No model-independent determination of  0 ! Progressive construction of P by connecting adjacent segment of Free parameter for each image  0  Y axis value fixed, X axis rescaled by a free factor

72. Construction of the EOS

73. Using our EOS : Trapped gas Previous works : EOS of the trapped gas (Duke, Boulder) Can be computed using our EOS + LDA ! L. Luo and J. Thomas, JLTP 154, 1-29 (2009) Legendre transform Integrals  Discrete sums over experimental data for h  Results independent of fitting or interpolation functions

74. Trapped gas : Results We find : Boulder Duke Tokyo Innsbruck

75. Imbalanced gas preparation - RF evaporation of 7 Li  cooling of 6 Li - Loading of 6 Li alone in the optical trap 10 µs |1>|2>... 10 µs |1>|2> Absorption picturesReference pictures - Fast imaging of the two spin states - Landau-Zener sweep for imbalanced spin mixture

76. Previous episode : Typical density profiles SF core Partially polarized normal phase Fully polarized shell S. Nascimbène, N.N, K. Jiang, L. Tarruell, M. Teichmann, J. McKeever, F. Chevy, C. Salomon PRL 103, 170402 (2009) Vertical width = 40 µm