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C. Salomon

Thermodynamics of a Tunable Fermi Gas. C. Salomon. Saclay, June 2, 2010. The ENS Fermi Gas Team. S. Nascimbène, N. Navon, K. Jiang, F. Chevy, C. S. L. Tarruell, M. Teichmann, J. McKeever, K. Magalh ã es,. Ridinger, T. Salez, S. Chaudhuri, U. Eismann, D. Wilkowski , F. Chevy,

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C. Salomon

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

  2. The ENS Fermi Gas Team S. Nascimbène,N. Navon, K. Jiang, F. Chevy, C. S. L. Tarruell, M. Teichmann, J. McKeever, K. Magalhães, • 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. Neutron star, Spin ½ G. Baym, J. Carlson, G. Bertsch,… The Equation of State of a Fermi Gas with Tunable Interactions Cold atoms, Spin ½ Dilute gas : 1014 at/cm3, T=100nK BEC-BCS crossover Spin imbalance, exotic phases TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

  4. EB=-h2/ma2 Lithium 6 Feshbach resonance

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

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

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

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

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

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

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

  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 Inverse of the fugacity : balanced gas at finite T, m1=m2=m : imbalanced gas at T=0

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

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

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

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

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

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

  20. Definition : dm=m1-m2 P: spin polarization What happens to superfluidity with imbalanced Fermi Spheres ? Superconductors: apply an external magnetic field but Meissner effect Cold Atoms: change spin populations 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 : balanced gas at finite T Inverse of the fugacity : imbalanced gas at T=0

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

  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 Schirotzek et. al, PRL 101 (2009) • Can we describe the normal phase as a Fermi liquid of polarons ? • binding energy of a polaron in the Fermi sea • effective mass

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

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

  26. Ground state of a tunable Fermi gas • Single-component Fermi gas: • Two-component Fermi gas δ1: grand-canonical analog of η: chemical potential imbalance

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

  28. Phase diagram BCS BEC

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

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

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

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

  33. Measurement of the Lee-Huang-Yang correction mean-field LHY BEC BCS 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)

  34. Beyond the Lee-Huang-Yang correction 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)

  35. Exp. 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) Direct Comparison to Many-Body Theories Grand-Canonical – Canonical Ensemble Fixed-Node Monte-Carlo theories Chang et al, PRA 70, 43602 (2004) Astrakharchik et al, PRL 93, 200404 (2004) Pilati et al, PRL 100, 030401 (2008)

  36. 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 • Next:Mapping the EOS in the complete space •  imbalanced gas at finite T , mass imbalance - Lattice experiments 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

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

  38. 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

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

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

  41. 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

  42. Trappedgas : Results We find : Boulder Tokyo Duke Innsbruck

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

  44. Simulation of Neutron Matter Neutron characteristics • spin ½ • scattering length • effective range Universal regime: • « dilute » limit  (mean density ) • Tc=1010 K =1 MeV, T=TF/100 kFa ~ -4,-10,…

  45. n(k) n(k) Grand potential 1 1 Chemical potential Pressure Temperature Atom number Volume Entropy Internal energy k/kF k/kF 1 1 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) 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,….

  46. MIT and Rice experiments MIT: M. Zwierlein, A. Schirotzek, C. Schunck, and W. Ketterle, Science 311, 492 (2006). Difference of optical density of the two spin populations RICE: G. Partridge, W. Li, R. Kamar, Y. Liao, and R. Hulet, Science 311, 503 (2006). 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

  47. 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) Direct comparison to many-body theories Grand-Canonical – Canonical Ensemble Fixed-Node Monte-Carlo theories Chang et al, PRA 70, 43602 (2004) Astrakharchik et al, PRL 93, 200404 (2004) Pilati et al, PRL 100, 030401 (2008)

  48. Polaron effective mass 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) Collective modes S.Nascimbene. et al, PRL 103, 170304 (2009) MIT measurement Y. Shin, PRA 77, 041603 (2008) We fit our data in the region

  49. Determination of m0 Density profile  segment of Unknown ! 7Li No model-independent determination of m0 ! Progressive construction of P by connecting adjacent segments From high temperature side assuming viriel 2 coefficient Free parameter for each image m0  Y axis value fixed, X axis rescaled by a free factor

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