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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, - PowerPoint PPT Presentation
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C. Salomon Saclay, June 2, 2010 Thermodynamics of a Tunable Fermi Gas
  • C. SalomonSaclay, June 2, 2010Thermodynamics of a Tunable Fermi Gas

  • The ENS Fermi Gas TeamS. Nascimbne, N. Navon, K. Jiang, F. Chevy, C. S.L. Tarruell, M. Teichmann, J. McKeever, K. Magalhes, Ridinger, T. Salez, S. Chaudhuri, U. Eismann, D. Wilkowski, F. Chevy,Y. Castin, M. Antezza, C. SalomonTheory collaborators: D. Petrov, G. Shlyapnikov , R. Papoular, J. Dalibard, R. Combescot, C. MoraC. Lobo, S. Stringari, I. Carusotto, L. Dao, O. Parcollet, C. Kollath, J.S. Bernier, L. De Leo, M. Khl, A. Georges

  • TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAThe Equation of State of a Fermi Gas with Tunable InteractionsCold atoms, Spin Dilute gas : 1014 at/cm3, T=100nKBEC-BCS crossoverSpin imbalance, exotic phases

  • Lithium 6 Feshbach resonance

  • No bound stateBound state

    Tuning interactions in Fermi gases Lithium 6 a>0a

  • Experimental sequence- Tune magnetic field to Feshbach resonance- Evaporation of 6Li - Image of 6Li in-situ

  • Spin balanced Unitary Fermi Gas

  • Direct proof of superfluidityMIT 2006Critical SF temperature = 0.19 TF

  • Thermodynamics of a Fermi gasVariables :scattering lengthatemperatureTchemical potential We build the dimensionless parameters : We have measured the EoS of the homogeneous Fermi gasInteraction parameterFugacity (inverse)Canonical analogsPressure contains all the thermodynamic information

  • Local density approximation:gas locally homogeneous at

    Measuring the EoS of the Homogeneous Gasi=1, spin upi=2, spin down

  • Extraction of the pressure from in situ images

    doubly-integrated density profilesequation of state measured for all values of

    Ho, T.L. & Zhou, Q.,Nature Physics, 09Measuring the EoS of the Homogeneous Gas

  • The Equation of State at unitarityThermodynamics is universalJ. Ho, E. Mueller, 04S. Nascimbene et al., Nature, 463, 1057, (Feb. 2010)

  • Phase diagram: exploring two fundamental sectors: balanced gas at finite T, m1=m2=m: imbalanced gas at T=0Inverse of the fugacity

  • Equation of state of balanced gas Accuracy: 6%High temperatureLow temperatureSuperfluid region

  • High T : virial expansionSF

  • Comparison with Many-Body Theories (1)

  • Comparison with Many-Body Theories (2)R. Combescot, Alzetta, Leyronas, PRA, 09

  • Low TemperatureA. Bulgac et al., PRL 99, 120401 (2006)R. Haussmann. et al., PRA 75, 023610 (2007)Normal phase : Landau theory of the Fermi liquid ?B. Svistunov, Prokofiev, 2006Normal phaseExp. data

  • Normal-Superfluid phase transitionWe find the critical parametersE. 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 liquidof quasiparticlesSuperfluidtransition(m/EF)c= 0.49 (2)

  • What happens to superfluidity with imbalanced Fermi Spheres ?Superconductors: apply an external magnetic fieldbut Meissner effect

    Cold Atoms: change spin populationsA question discussed extensively since the BCS theory and more than 30 papers in the last 3 yearsMIT 06,: 3 phases, RICE 06: 2 phases, ENS 09: 3 phases

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

  • Equation of state h(h, 0) i.e.(T=0)Deviation from hs atT=0 SF-Normal Phase TransitionMIT: Y. Shin, PRA 08 Fixed-NodeSFMixed normal phaseIdeal gas

  • An interesting limit: the Fermi PolaronPartially 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)

  • Ideal gas of polaronsEOS for a Fermi liquid of polaronsFixed Node (Lobo et. al.)Mixed normal phase:Ideal gas + ideal gas of polarons

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

  • Single-component Fermi gas:

    Two-component Fermi gas

    1: grand-canonical analog of : chemical potential imbalance Ground state of a tunable Fermi gas

  • Equations of State in the Crossover First order phase transitions: slope of h is discontinuousPairedSF polarizedNormal phaseIdeal F GasPairedSF

  • Phase diagramBCSBEC

  • SFSFWe use the most advanced calculations forProkofev et al., R. Combescot et al, BEC side, unitarity: excellent agreementBCS side: deviation close to c

    Comparison with the two ideal gases model

  • Full pairing:

    Symmetric parametrization: Superfluid Equation of State

  • Superfluid Equation of State in the Crossover BCSBEC1/a= 0

  • BCS limit:

    BEC limit

    Unitary limitLee-Yang correctionmean-fieldmolecularbinding energymean-fieldLee-Huang-Yang correctionWe get: xs= 0.41(1)contact coefficient z= 0.93(5)Asymptotic behaviors

  • Fit of the LHY coefficient: 4.4(5)


    No effect of the composite nature of the dimersX. Leyronas et al, PRL 99, 170402 (2007)

    mean-fieldLHYMeasurement of the Lee-Huang-Yang correctionBCSBEC

  • For elementary bosons:

    B is not universal for elementary bosons (Efimov physics)

    *: three-body parameterE. Braaten et al, PRL 88, 040401 (2002)

    Here: universal value (using an appropriate Pad fit function)

    Beyond the Lee-Huang-Yang correction

  • Direct Comparison to Many-Body TheoriesGrand-Canonical Canonical EnsembleChang 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

  • Conclusion - Perspectives EOS of a uniform Fermi gas at unitarity in two sectors1) balanced gas at finite T2) T = 0 imbalanced gas Precision Test of Many-body Theories

  • Thermodynamics of a unitary Fermi gas- Equation of state of a homogeneous two-component Fermi gas (appropriate variables experimentally) immersing weakly interacting bosonic 7Li (ideal thermometer)

  • ConclusionMeasurement of the EoS :- Unitary gas at finite temperature- Fermi gas T=0 in the BEC-BCS crossoverQuantitative many-body physics benchmark for theories

  • Comparison with Tokyo groupGrand-canonicalCanonical ensemble Viriel 2?Disagrees with Viriel 2 expansionM. Horikoshi, et al. Science 327, 442 (2010);TokyoENS

  • Typical imagesOne experimental run : density profile + temperature - Image of 6Li in-situImage of 7Li in TOF

  • ConclusionMeasurement of the EoS :- Unitary gas at finite temperature- Fermi gas T=0 in the BEC-BCS crossoverQuantitative many-body physics benchmark for theories

  • Trapped gas : Results

  • Perspectives Open Questions- Critical temperature in the BEC-BCS crossover- Nature of the Normal phase in the crossover- Low-lying excitations of the superfluidFermi liquidPseudo-gapSuperfluid

  • Neutron characteristicsspin scattering lengtheffective range

    Universal regime: dilute limit (mean density )Tc=1010 K =1 MeV, T=TF/100 kFa ~ -4,-10,

    Simulation of Neutron Matter

  • ThermodynamicsIs a useful but incomplete equation of state !Complete information is given by thermodynamic potentials: Equ. of state useful for engines, chemistry, phase transitions,.

  • MIT and Rice experimentsObvious phase separation seen on the in situ optical density differenceClogston limit at MIT is P= 0.75 and close to 1 at RiceMIT: 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).Differenceof optical density of the two spinpopulations

  • Direct comparison to many-body theoriesGrand-Canonical Canonical EnsembleChang 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

  • We fit our data in the region

    Simple analytical theoryR. Combescot et al, PRL 98, 180402 (2007)

    Fixed Node Monte CarloS. Pilati et al, PRL 100, 030401 (2008)

    Diagrammatic Monte CarloN. Prokofev et al, PRB 77, 020408 (2008)

    Most advanced analytical theoryR. Combescot et al, EPL 88, 60007 (2009)

    Polaron effective mass Collective modesS.Nascimbene. et al, PRL 103, 170304 (2009)MIT measurementY. Shin, PRA 77, 041603 (2008)

  • Determination of m0Density profile segment of

  • Construction of the EOS

  • superfluid core normal phase

    Superfluid core for

    We calculate using fits of our EoS Application: Critical polarization for superfluidity

  • Partially polarized phaseSimple physical picture: one builds a Fermi sea of fermionic polaronsSingle-polaron ground state:Excited states:


    Fermi pressure of majority atomsFermi pressure of polarons

  • Using 7Li as a thermometer 6Li 7Li - Feshbach resonance @ 834 Ga76 = 2 nm Good ! Weakly interactinga77 = -3 nm Not so good BEC instabilityclose to expected Tc for SF transition

  • Clogston-Chandrasekhar limit Naive argument using BCS picture :the energy of excess particles must be compared with robustness of the fermion pairs :

  • 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

  • Previous works: superfluid equation of stateRadial breathing modeexperimentsAltmeyer et al, PRL 98, 040401 (2006)

    theoryMean fieldQMC calculations +hydrodynamics +scaling ansatzAstrakharchik et al, PRL 95, 030404 (2005)

  • Extracting the BCS asymptotic behaviorOn the BCS side: Pade approximant

    mean-field value used as a constraint

    From :

    Lee-Yang coefficient: 0.18(2)


    mean-fieldmean-field + LY

  • Using and , one calculates

    at : jump of n2/n1

    low imbalance phase: n2=n1

    large imbalance phase: 0

  • 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

  • -- 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-bodysystems. -- In progress: General equation of state at finite temperature (ongoing !) Superfluid temperature = f(kFa), polaron oscillation (PRL 09)-- FFLO phase,-- Lattice experiments and low D systems


  • A good reading !Enrico Fermi on lake Como

  • Experimental approach Cooling of 7Li and 6Li

    1000 K: oven

    1 mK: laser cooling

    10 mK: evaporative cooling in magnetic trap

    E= - m.B Ioffe-Pritchardtrap

  • 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, 6Li40K 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 physicsQuantum simulation with cold atomsPeriod: 4.9 mmD. Weiss et al., Nature Phys, 2007

  • Dispersion relation of polaronsLow 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, MCN. Nascimbene, N. Navon, K. Jiang, L. Tarruell, J. Mc Keever, M. Teichmann, F. Chevy, C. Salomon, to appear in PRL 2009 arXiv:0907.3032

  • Determination of xUniversal equation of state of the unitary Fermi gas at zero temperatureForx is a universal number

    ExperimentENS (6Li)0.42(15)Rice (6Li)0.46(5)JILA(40K)0.46(10)Innsbruck (6Li)0.27(10)Duke (6Li)0.51(4)


  • Double-integrated profiles of density differenceMITFlat top = hollow core with LDA = paired coreEffect of the number of atoms, aspect ratio and temperature ?

  • In-situ density profiles easy to interpret at T=0 Trapping potential is useful (this time !)Theory, F. Chevy 06, Lobo et al.,

  • Typical imagesOne experimental run : we get density profile + temperature Density profile of 6LiTemperature 6Li7LiMixture remarkably stable !

  • Determination of m0Density profile segment of

  • Construction of the EOS

  • Using our EOS : Trapped gasPrevious 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)

  • Trapped gas : Results

  • Imbalanced gas preparation- RF evaporation of 7Li cooling of 6Li - Loading of 6Li alone in the optical trap

  • Previous episode : Typical density profilesS. Nascimbne, N.N, K. Jiang, L. Tarruell, M. Teichmann, J. McKeever, F. Chevy, C. Salomon PRL 103, 170402 (2009)

    First of all I would like to thank the organizers for this very interesting workshop, bringing together physicists from different fields, leading ato a very contsruvtive approach. I would like to present new results on the thermodynamics of ultracold Fermi gases. Take a system of fermions with two spin states. At low temperature they interact only trough s wave interaction. This is the simplest many-body system! Yet the phase diagram is rather rich and very hard to compute in the strongly interacting regime. We have used a new method to measure the equation of State of ultracold gases which enables one to directly measure the pressure of the gas, hence the grand potential in the grand canonical ensemble. This quantity contains all thermodynamic properties such as energy, entropie or chemical potential, phase transitions, and so forth. The precise equation of state is used to deduce some new properties of strongly interacting gases in the crossover between BEC and BCS. And to make adetalied comparison with advanced theories. This comparison has revealed some surprises !The beautiful beaches and sea in Rio made me choose this title even if the temperature is 10 orders of magnitude lower than the sea here ! In this tak I would like to describe recent experiments at ENS on cold Fermi gases and develop the analogy with condensed matter system and high Tc superfluids. I will focus on the case of strong attraction and discuss the thermodynamic properties of a strongly interacting Fermi gas. phase diagram at zero temperature

    Citer Frederci Chevy.A lot has been done ! But comparison with theory and the determination of the equation of state was lacking because of the effect of the harmonic trap. In the trap the density is inhomogeneous and may quatities are averaged over the trap. I will show how to turn this inhomogeneity to an advantage !*On vapore le 7Li dans un pige magntique, qui refroidit sympathiquement le Li6, puis, on charge le restant du mlange dans un pige optique o on applique le champ magntique de Feshbach. On effectue une vaporation du Li6 en interaction forte (qui refroidit son tour le Li7) et on effectue finalement une image des deux isotopes, suivant cet axe pour avoir le profil axial du Li6 et suivant cet axe en temps de vol pour le Li7 pour obtenir sa temprature avec une densit optique plus leve*Temperature is not easily measured on this system. Is the system SF ? What can you deduce from such in situ image ? In fact a lot ! I will show you that by careful analysis of such images the full equation of state of the cold gas can be measured !!including phase transition and the study of the normal phase ! *Our goal is to measure the EoS of the homogeneous Fermi gas, where the pressure is simply related to the grand potential. So, the pressure contains all the thermodynamic information on the system. We have three relevant variables from which we can build two dimensionless parameters, an interaction parameter delta, using mu and a and the inverse of the fugacity zeta. Notice that in the more usual language of the canonical ensemble, delta is analog to 1/kFa and zeta is analogous to T/TF. *** la fonction P est rapporte la pression dun gaz sans interaction une composante P_1 qui est une fonction poylog dordre 5/2. On obtient une fonction h qui ne dpend que de deux variables, eta le rapport des potentiels chimiques, et zeta linverse de la fugacit. Nous nous sommes concentrs sur deux secteurs, le gaz quil T fini, et le gaz dsquilibr temprature nulle.Indiquer high temperature, low temperature. Pression normalise par celle du gaz sans interaction de Fermi la mme temperature et au mme potentiel chimiqueThe low temperature data can be also plotted as a function of T/mu square. It idsplays a remarquable linear slope caracterisitc of Fermi liquid theory. We can then extract a number of new parameters from ths theory as the effective mass of quasi-particles and the compressibility of the gas at zero temperature xsi_n. the effective mass of the quasi-partciles is quite close to the single fermion mass of 1, a remarquable fact considering the strong interactions This is in strong contrats with helium 3 where the pairing is different P-wave but the effective mass ranges from 6 to 30 depending on pressure !

    Let us focus then on the zero temperature limit. t on the zero temperature axis, the vertical axis. Our data points stop at a value remarquably close to the blue dot-dashed line with xsi at the power -3/2. This is the zero temperature pressure and it has been calculated by fixed node Monte-Carlo method and diagrammatic expansions. *Pour uniformiser la suite, on adimmensionne la fonction P la pression dun gaz sans interaction une composante P_1 qui est une fonction poylog dordre 5/2. On obtient une fonction h qui ne dpend que de deux variables, eta le rapport des potentiels chimiques, et zeta linverse de la fugacit. Nous nous sommes concentrs sur deux secteurs, le gaz quil T fini, et le gaz dsquilibr temprature nulle.*Now I would like to concentrate on the intermediate phase: non SF but strongly interacting.************Toutes les proprits thermodynamiques dun systme sont contenues dans ses potentiels thermodynamiques. En particulier, dans lensemble grand-canonique, le grand-potentiel dpend des potentiels chimiques des deux espces de spin, de la temprature et du volume et scrit simplement comme le produit pV o p est la pression du gaz. Jutilise cet ensemble car il sagit des variables pertinentes dans la suite. La mesure dune telle quation dtat a un intrt norme, dautant plus dans un systme en interaction forte comme les gaz de fermions deux composantes, qui possdent des proprits universelles. La raison de cette absence est simple et se rsume deux problmes principaux. Dune part, la thermomtrie dun gaz fortement corrl est trs dlicate car les profils de densits in-situ ou en temps de vol sont en gnral inconnus mais le problme le plus srieux provient de la prsence dun potentiel de confinement, qui rend le gaz inhomogne. Les mesures reprsentent alors une moyenne des grandeurs sur le pige. Si je suis ici, cest parce que nous avons pu surmonter ces deux difficults. La premire, en immergeant dans le gaz fermionique, un nuage de Li7 en interaction faible, qui fournit une thermomtrie directe du gaz de fermions.We first image Li6 in-situ, for the pressure measurement. Then, we switch off the trap and measure along the long-axis the Li7 in time of flight. So, in a single experimental run, we obtain the density profile and the temperature. Here I show you typical images, of 25000 fermionic atoms, with around the same number of Li7 at a rather high temperature of 1.2 microK.**Limit of largescattering length: universal physics2) Dynamically excite the gas by changing a.******The imbalance limit at which the superfluidity breaks down is known as the CC-limit. We can try to have a naive understanding of this limit. First, in the BCS limit of weakly interacting fermions, we must compare the energy of excess particles with the robustness of the fermion pairs. This is just comparing the difference of chemical potentials with the pairing gap parameter. If this difference is smaller than the gap, the pairs survive and the superfluid phase is stable. If it is not, the additional particles will break the pairs and destroy the superfluid. Notice that this happens because the pairing energy depends on the other particles.In the deep BEC side, the picture is even simpler because the gas is constituted of tighly bound dimers. They are weakly interacting and their binding energy does not depend upon other particles. We then expect that these molecules will not be affected by additional atoms. The ground state will be a mBEC immersed in an ideal gas of excess atoms. The BCS theory provides a quantative criterion for the CC-limit, that is the following one. I introduce now an important quantity, the polarization P. The value 0 corresponds to a balanced gas, the value of 1, is the fully polarized gas. The Rice data remains a mystery ??? Out of equilibrium situation ? ****Now I would like to take you on a small cruise on the Fermi sea . When we saw this picture of Enrico Fermi on the beautiful lake Como, with Massimo Inguscio and W. Ketterle we could not resist including it in the proceeedings of the E. Fermi school tat was held in 2006 on uktracold Fermi gases. The connection to the Fermi sea is then obvius !It was a real pleasure to collaborate with Antoine and his group on some of these methods.*How to interpret in situ ensity profiles ? What do we expect ?Assume T=0Rmemeber the trapping potential provides a spatially varying density, therefore a spatially varying chemical potentials, mu1 and Mu2. Hence of the ratio M1/Mu2. The gas wants to maximize its pressure and P is a monotonic function of the ratio heta.**