FRÉDÉRIC CHEVY 6ème Demi-Journée des Théoriciens

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1 Des gaz d'atomes ultra-froids pour comprendre la matière condensée FRÉDÉRIC CHEVY 6ème Demi-Journée des Théoriciens

2 ENS FERMI GROUPS S. Nascimbène N. Navon L. Tarruell FC K. Magalhaes C. Salomon S. Chaudhuri A. Ridinger T. Salez D. Wilkowsk U. Eismann NEW MEMBERS: B. Rem, I. Ferrier-Barbut, A. Grier, F. Sievers, D. Rio Fernandes, N. Kretzschmar COLLABORATIONS: Y. Castin, F. Werner, C. Mora, R. Combescot, X. Leyronas, W. Krauth, S. Piatecki, A. Georges, S. Giorgini, A. Recati, S. Stringari, C. Lobo, O. Goulko, O. Dulieu, L. Pruvost

3 R=3 Analog simulation? Solve the equation 3x=4 Analytics Numerics R.P. Feynman Analog simulation x=4/3 double x1=-2.0; double x2=2.0; Double epsilon=0.001; while(x2-x1>epsilon) { xm=(x1+x2)/2; if(3*xm>4) {x2=xm;} else{x1=xm;} }; I V 0 =4V A Ohm s Law, V 0 =RI 3I=4

4 Classical Simulators Antikytheron Mechanism (~-100), T. breeth et al., Nature (2006) Strasbourg s Astronomical Clock (~1550) Analog Computer SEA OME P2 (~1960)

5 A CENTURY OF SUPERCONDUCTIVITY DISCOVERY OF SUPERCONDUCTIVITY (1911) BCS IN A NUTSHELL (1957): Superconductivity arises from pairing of weakly interacting (identical) electrons with opposite spins WHAT IF? Interactions are strongly attractive (BEC/BCS crossover)? Spin populations are imbalanced (Clogston-Chandraseckhar limit)? Partners of the Cooper pair are different? Dimensionality is modified (1D-2D)

6 THE ENS LITHIUM EXPERIMENT 6 Li Feshbach resonance Bose-Einstein Condensate ( 7 Li) Fermi sea ( 6 Li) Atom number ~ Temperature ~100nK (T/T F 0.05)

7 FERMIONIC SUPERFLUIDITY V 0 V 0 DEEP POTENTIAL : stable 2-body bound state with binding energy E b =- 2 /ma 2. Many-body ground state: BEC of molecules (a>0) SHALLOW POTENTIAL (V 0 <V 0*, a<0): No 2-body bound state. No superfluid? BOSE ENSTEIN CONDENSATION AND BCS THEORY APPEAR AS TWO LIMITING CASES OF A UNIFYING THEORY: THE BEC-BCS CROSSOVER (Nozières, Schmitt-Rink, Legget)

8 Température critique THE BEC-BCS PATCHWORK Ultra-cold fermions BEC d alcalins Helium superfluide Supraconducteurs haute Tc He-3 superfluide Supraconducteurs métalliques Interactions EXPERIMENTAL REALIZATION IN COLD ATOMS (2003) : Innsbruck, JILA, MIT, ENS, Rice, Duke, Melbourne, Tokyo

9 The BEC-BCS crossover with cold atoms (MIT)

10 LOCAL DENSITY APPROXIMATION Measurement of the equation of state of a Fermi gas (T.L. Ho & Q. Zhou, Nature µ z µ 0 Physics 6, 131 (2009)) z Gibbs-Duhem 2 m r P ( µ, µ, T) n ( z) n ( z) z z 2 with «P» n iz i m z z /2 Each picture provides a realization of a piece of the equation of state. z «µ»

11 SPIN BALANCED GAS : THE BEC-BCS CROSSOVER P/P 0 MF: Mean field theory LH: Lee-Yang beyond mean field correction LHY: Lee-Huang-Yang beyond mean field corrections, with a dd ~0.6a (Petrov et al.) / a 2m

12 UNIVERSALITY OF BOSONIC BEYOND MEAN- FIELD CORRECTIONS We repeat the procedure for 7 Li bosonic atoms Mean Field Lee-Huang-Yang We find : 4.5(7) Averaging images between 1440 and 2150 a 0 N. Navon et al. arxiv:

13 SPIN IMBALANCED GAS : CLOGSTON- CHANDRASECKHAR LIMIT (a= ) P/P 0 P ni µ i h=m /m See also: Zwierlein et al. Science 311 (5760), (2006);G.B.Partridge et al., Science, 311 (2006) 503; for a review: Mora & Chevy, Prog. Rep. Phys (2010)

14 SWIMMING IN THE FERMI SEA POLARON : electrons immersed in a bath of phonons (bosons) FERMI POLARON: electron immersed in a Fermi Sea QUASI-PARTICULE PROPERTIES : (F.Chevy, PRA 74, (2006); R. Combescot et al. PRL 98, (2007);) p E( p ) µ p... * 2m a µ AE 0.6E m 1.17m p F F *

15 EXPERIMENTAL STUDY OF THE FERMI POLARON 2 p E( p) ~ p... * 2m RF SPECTROSCOPY (MIT, A. Schirotzeck et al. PRL 102, (2009)) COLLECTIVE EXCITATIONS (ENS, S. Nascimbène et al. PRL 103, (2009)) n rf MIT Experiment Quantum Monte Carlo Variational Theory Single atom µ p atom+ background m*=1.17(10)m ( N N ) / ( N N ) (C. Lobo et al., Phys. Rev. Lett. 97, (2006))

16 COLLECTIVE BEHAVIOR NORMAL PHASE : Ideal gas of polarons (FERMI LIQUID) Monte Carlo: C. Lobo et al., Phys. Rev. Lett. 97, (2006) Analytical : C. Mora et F. Chevy, Phys. Rev. Lett., Phys. Rev. Lett. 104, (2010). P /2 * 3/2 m 5/2 m 5/2 7/ p O p

17 EXPERIMENT 6 LI- 40 K (A. RIDINGER ET AL., arxiv: , accepted in EPJD) OBJECTIVE : Study of mass imbalanced Fermi mixtures (see also Innsbruck, Amsterdam, Munich/Singapore, MIT). Li K Performances: ~10 9 atoms of each species loaded in 5s PHOTOASSOCIATION OF LiK MOLECULES K 2 LiK

18 MIXED DIMENSIONALITY Y. Nshida Phys. Rev. A, (2010) Lamporesi et al., Phys. Rev. Lett. 104, (2010)

19 ! CONCLUSION AND OUTLOOK Finite temperature phase diagram (NB: T c /T F ~0.15 «HIGH-T C» SUPERCONDUCTIVITY) Nature of the Normal phase: Fermi Liquid vs non-fermi Liquid (ENS vs JILA) Fermions in optical lattices, disorder Superfluidity in mass imbalanced mixtures Heteronuclear molecules with long range dipolar interactions

20 Modèles à N-corps sur réseau CLASSIQUE ~C N possibilités {,,..., } C Cases N pions QUANTIQUE : Espace de Hilbert de dimension ~C N C1 C2 C N C1 C2,,..., CN

21 La simulation analogique? Cherchons à résoudre y (t)=-y(t) with y(0)=1 Le génie Calcul numérique Simulation analogique y(t)=e -t double t=1.0; double epsilon=0.001; int n=math.round(t/epsilon); double[] y=new double[n]; y[0]=1.0; for(int i=1,i<n,i++) { y[i]=y[i-1]-epsilon*y[i]; }; R C y(t) V 0 V 0 =1V; RC=1s;

22 Les simulateurs classiques Mécanisme d Anticythère (~-100), T. breeth et al., Nature (2006) L horloge astronomique de Strasbourg (~1550) Ordinateur Analogique SEA OME P2 (~1960)

23 The equation of state of an ultra-cold Fermi gas JILA, Innsbruck, MIT, ENS, Rice, Duke, Tokyo, Melbourne, Munich, Zurich Interactions courte portée, modulation de la longueur de diffusion (résonance de Feshbach) Contrôle des populations de spin Atomes piégés (!) POTENTIELS ARBITRAIRES (réseaux optiques) APPROXIMATION DE DENSITE LOCALE with iz i m z z /2 (T.L. Ho & Q. Zhou, Nature Physics 6, 131 (2009)

24 THE TRAP BLESSING AND THE EQUATION OF STATE OF AN ULTRA-COLD GAS ARBITRARY POTENTIALS (optical lattices, reduced dimension) LOCAL DENSITY APPROXIMATION: the gas behaves locally as an homogeneous system and provides a realization of a piece of the equation of state. Problem: how do you deconvolve the line of sight integration to get access to local information? Abel transform (Y. Shin, PRA 77, (2008)) «Magic formula» (T.L. Ho & Q. Zhou, Nature Physics 6, 131 (2009)): you do not need to deconvolve! with iz i m z z /2