Ultra-cold Fermi Gases From Molecular Bose-Einstein Condensation to BCS Pairing Ultra-cold Fermi Gases From Molecular ose-Einstein Condensation to BCS Pairing Thomas Bourdel, Julien Cubizolles, K. Magalhaes Servaas Kokkelmans, Dima Petrov, Gora Shlyapnikov Christophe Salomon Laboratoire Kastler Brossel, Ecole Normale Supérieure, Paris, Orsay, 20 novembre 2003 Collège de France Molecular Bose-Einstein Condensate and Fermi Superfluidity Molecular Bose-Einstein Condensate and Fermi Superfluidity Two component Fermi gas at very low temperature s-wave interaction, scattering length a a > 0 a < 0 Fermi superfluid ?Molecular BEC ? See Science news focus, August 8th, 2003 Feshbach resonance Outline Outline Basics of dilute trapped Bose and Fermi gases Study of 6 Li Feshbach resonance - Measurement of the interaction energy near a Feshbach resonance - Fermions in the strongly interacting regime Search for superfluidity in Fermi gas Formation of ultra-cold long-lived molecules near Feshbach resonance Reversible process between Fermi and Bose gases 40 K 2 , 6 Li 2 dimer molecules Bose-Einstein condensation of molecules Perspectives Bose -Einstein Condensation in Atomic Gases Bose -Einstein Condensation in Atomic Gases Bose-Einstein condensate Rubidium Bose enhancement MI T T>T c T<T c T<<T c Sodium (0.83 N) 1/3 = T C k B hω Rb, Na, Li, H, *He, K, Cs, Yb, K 2 , Li 2: dimers of fermions 1995 E. Cornell /C. Wieman, W. Ketterle Prix Nobel de physique 1997 S. Chu, C. Cohen Tannoudji, W. Phillips Manipulation d’atomes par laser Prix Nobel de physique 2001 E. Cornell, W. Ketterle, C. Wieman Condensation de Bose-Einstein Quantum Statistics Quantum Statistics 3 He, electrons in metals, atoms, neutron stars, … Fermi-Dirac statistics (1926) E F Fermi sea Pauli Exclusion T << T = (6 N) F 1/3 k B hω Bose-Einstein statistics (1924) Bose-Einstein condensate Bose enhancement (0.83 N) 1/3 = T C k B hω superfluid 4 He, dilute gases, excitons 22 1 () 2 Vr m rω= null Gaseous Condensates : orders of magnitude Gaseous Condensates : orders of magnitude Dilute gaz at temperature T confined in harmonic trap : Condensation threshold: n 0 : central density 3 1.202 B kT N ω ?? = ?? ?? null 3 0 2.612n λ = B kT ωnullnull 2 B h mk T λ π = Liquid Helium : 10 27 atoms/m 3 n 0 -1/3 = 10 ? T ~1 K Gaseous condensates: 10 19 atoms/m 3 n 0 -1/3 = 0.5 μmT ~1 μK Magnetic trapping of neutral atoms Magnetic trapping of neutral atoms BBE nullnull μμ +=?= . nullnull local minimum of B null + spin polarization Trap depth : 1 mK The magnetic energy creates a potential well for the center of mass motion of the atoms For loading a magnetic trap: Laser cooling : nλ 3 ≈10 ?6 Photo: Bell Labs 10 9 atomes, 1 cm 3 100 μK mélasse optique Mesure in situ: distribution en position ou après temps de vol: distribution en impulsion Interest of dilute Bose and Fermi gases Interest of dilute Bose and Fermi gases 1) Low density, low energy 10 12 -10 15 at/cm 3 , T~ 10 -10 eV~1 μK. E Fermi ~ 10 μK Atom-atom interactions described by a small number of parameters Scattering length, density,..tunability of interactions Two-body, three-body interactions Flexibility of trapping parameters 2) Simplicity of detection by optical imaging Comparison between experiments and predictions of many-body theories Gross-Pitaevski eq., Bose-Hubbard model, Mott insulator transition,… Link with other fields of physics, condensed matter, solid-state physics nuclear physics, astrophysics,…. Fermi systems Fermi pressure Inhibition of collisions, Modification of spontaneous emission rate Mixtures of Bose-Fermi systems Search for a BCS transition Atom-atom interactions Atom-atom interactions The magnitude and sign of a depend sensitively on the detailed shape of long range potential r U(r) At low temperature, only s wave collisions . 0 0 () tan ( ) lim ik r ikr k a re e r k a k ψ δ → =? =? null null null a: scattering length |a|~1 à 100 nm 2 12 12 4 () () a Vr r r r m π δ? =? null nullnullnullnull a > 0 : effective repulsive interaction a < 0 : effective attractive interaction 2 2 () () () () 2 Vr Ng r r r m ψψμψ ?? ??+ + = ?? ?? null nullnullnullnull a : scattering length 2 4 a g m π = null Condensate dynamics described by Gross-Pitaevski equation : scattering scattering length length 0 0.02 0.04-0.04 -0.02 6 6 C C ? Control of interactions in a condensate Control of interactions in a condensate W(r) r 6 6 /Cr? |a| varies typically between 1 and 100 nanometers (Bohr radius : 0,05 nm) The scattering length: a Scattering cross section: σ = 4 π a 2 (for non identical particles) Example: BEC in dependence of mean field Soliton 1D gas At very low temperatures one parameter is sufficient to describe atomic interactions: the scattering length a Parabola Repulsive: a>0 Collapse (for N>N crit ) 3D gas Ideal gas: a=0 Gaussian Mean field of a gas with density n(r) U(r) = m 4 π h 2 n(r) a NL GPE The scattering length: a Attractive: a<0 Evaporative cooling Evaporative cooling Evaporative cooling is today the only known method for reaching quantum degeneracy : - Remove hot atoms - Use elastic collisions for rethermalization Cooling of fermions: sympathetic cooling Cooling of fermions: sympathetic cooling Evaporative cooling: Only known method for reaching quantum degeneracy Problem: anti- symetrization for fermions Solution: Sympathetic cooling of distinguishable particles: - Fermions in two different spin states : 40 K JILA, 6 Li Duke - Fermions-bosons: two isotopes of the same atom: 7 Li , 6 Li , ENS, Rice - Fermions-bosons: differents elements: 40 K, 87 Rb, LENS, JILA 23 Na, 6 Li MIT - Current Fermi degeneracy T~ 0.05 T F for 10 6 to 10 7 fermions (MIT, JILA, 2003) Will this be sufficient for BCS pairing ? Bose Einstein condensate and Fermi sea Bose Einstein condensate and Fermi sea Lithium 7 Lithium 6 T bosons =T fermions = 0.2 T F F. Schreck, L. Khaykovich, K. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, C. Salomon PRL, 87, 080403, 2001 Search for superfluidity in Fermi gas Formation of Cooper pairs Search for superfluidity in Fermi gas Formation of Cooper pairs Bardeen, Cooper, Schrieffer, 1957 Superconductivity in metals at low temperature: Simple picture:Homogeneous degenerate Fermi gas, k F , E F Add two fermions, 1 et 2, with attractive interaction: < 0 with V < 0 Then, these particles can always form a state with an energy lower than E F , a bound state. Pairs at Fermi surface: If the temperature is low enough, these pairs form a superfluid phase 12 12 ()()Vr r V r rδ?= ? nullnull nullnull ,kk? nullnull || F kk≥ 2|| 0.3 F ka BCS F TTe π ↑↓ ? ～ 1 F ka<< a ↑↓ Near Feshbach resonance: T BCS ~ 0.26 --0.5 T F, Holland et al., In optical lattice: T BCS ~ 0.05 T F , Hofstetter et al. for ? Coupling between open channel and closed channel Closed channel Open channel a B a bg E kin magnetic field Feshbach Resonance Feshbach Resonance Resonance: short range molecular state What is the lifetime of vibrational levels close to dissociation limit ? Scattering is strongly energy dependent Lithium 6 Feshbach resonance Lithium 6 Feshbach resonance a = + 0.27 nm μ μ= 1/2 b { |3/2,+3/2> |1/2,-1/2> |1/2,+1/2> a = + 40 a o 6,7 6,7 a = + 38 a o Lithium 6 27 G B μ μ= 1 b μ μ= 1/3 b { { -110 nm Interesting region for molecule formation interesting region for BCS 0,0 0,5 1,0 1,5 2,0 -200 -100 0 100 200 scattering length [nm] Magnetic field [kG] Lithium Magneto-optical Trap Lithium Magneto-optical Trap M.O. Mewes et al., PR A 61, 011403R (2001) Strongly interacting 6 Li gas in an optical trap Strongly interacting 6 Li gas in an optical trap T F = 5 μK T/T F = 0.22 N total = 1 10 5 E interaction = -0.35 E kin, k F |a| > 1 na 3 > 1 Duke, ENS, MIT Two YAG beams with 2.5 W and waist of 38 μm Measurement of the interaction energy Measurement of the interaction energy Energy of the trapped gas x y z 350 mμ a b intkintot l pota E EE E=++ kr in E E= Time of flight images a) Expansion without magnetic field b) Expansion with magnetic field intkinr E EE=+ E int < 0 Expansion is governed by collisional hydrodynamics T. Bourdel et al., ENS, PRL, 91, 2003 Interaction energy vs magnetic field: surprises Interaction energy vs magnetic field: surprises 600 700 800 900 1000 -0.4 -0.2 0.0 0.2 0.4 0.6 E int /E kin Magnetic field [G] Effect of molecules ? Resonance found at 810(20) Gauss in good agreement with theory Interaction energy is negative on resonance, as predicted by Heiselberg Change of sign of E int is shifted from resonance Strongly interacting Fermi gas: k F |a| > 1 Loss region k F a=0.5 at 720 G Molecules in the system ? Molecules in the system ? ? Shift of resonance? B peak = 855 +- 53 Gauss unlikely! ? Three-body recombination [D. Petrov, PRA 67, 010703 (2003)] – Molecules form efficiently in highest weakly bound state 2 2 ma E B null = Molecules can be trapped! +E B Binding energy released 1 () ( 2 ) exp( / )rra ra?π ? =? E B < E trap Particles stay in trap E B > E trap Trap loss Molecule formation: time dependent process Molecule formation: time dependent process Proposed for bosons in 2000-2001: Timmermans et al.,Verhaar et al., Julienne, Burnett et al. B o Short range molecular bound state a < 0 atoms a > 0 atoms 22 / B Ema=?null B E kin When crossing the resonance from right to left, it is energetically favorable to form weakly bound molecules with binding energy E B which remain in the dipole trap If slow enough, adiabatic and reversible process: entropy is conserved Recent experiments Recent experiments With bosons: 85 Rb, JILA, Donley et al, 2002: coherent oscillations between a BEC and molecules 133 Cs, Innsbruck, Herbig et al, 2003: direct imaging of the molecules which separate from BEC via Stern and Gerlach expt 133 Cs, Stanford, Chin et al., 2003: spectroscopic detection of molecules 87 Rb, MPQ, Dürr et al., 2003: 1D trapping of molecules But lifetime limitations: observed losses. Work at low density With fermions: Long lifetime ! 40 K, JILA, Regal et al., 2003: direct imaging via RF dissociation and Stern and Gerlach separation, measurement of binding energy and lifetime 6 Li, ENS, 2003, Cubizolles et al., 2003: observation of long lifetime (0.5 s); conversion efficiency 85%, T/T BEC ~ 2 6 Li, Rice, Strecker et al., 2003: long lifetime 6 Li, Innsbruck, Selim et al., 2003: pure trapped molecular cloud, long lifetime 40 K, JILA, Regal et al., 2003: BEC of K 2 dimers 6 Li, Innsbruck, Selim et al., 2003: BEC of Li 2 dimers Time-dependent experiment at ENS: reversible formation of ultracold Li 2 molecules Time-dependent experiment at ENS: reversible formation of ultracold Li 2 molecules 0,0 0,5 1,0 1,5 2,0 -200 -100 0 100 200 s c a tter ing length [nm] Magnetic field [kG] 1 2 4 N 1 = 8 10 4 atoms initially @ (1) Change magnetic field in 50 ms towards (2), where a is >0 and large - Count atom number N 2 @ (2) Return to 1 in 50 ms and count N round-trip =N 3 -Detection @ (4) Abruptly switch-off B field in 20 μs @ B=0, a=0 Only free atoms are detected: Two components: atoms and broken molecules Importance of dB/dt Binding energy of molecule 22 / B Ema=null 3 Efficiency of molecule formation influence of trap depth Efficiency of molecule formation influence of trap depth At peak : 3 10 4 cold bosonic molecules Most efficient when: T/T F is small and T/E B is small. The colder the better ! At B= 689 G a = 78 nm E B = 12 μK At peak: U trap /k B = 10 μK ω=2π 1.4 kHz T F = 5 μK T= 6.7 μK n 0m ～4 10 13 molec/cm 3 Critical temperature for molecules:T C = 3.5 μK Factor 2 to gain for BEC of molecules Temperature of atom-molecule mixture Temperature of atom-molecule mixture T atoms at 2 T atoms at 3 resonance Atoms Molecules : heating The molecule binding energy is transferred to external motion Molecules atoms : cooling Reverse process Very little heating after 100 ms round-trip reversible process: entropy is nearly conserved Critical temperature for molecules: T C = 3.5 μK Factor 2 to gain for BEC of molecules Dimers of fermions live very long Strong dependence upon a Dimers of fermions live very long Strong dependence upon a a = 78 nm a = 35 nm The larger is a, the longer the lifetime Role of Fermi statistics G~ 1/a s with s = 2.55 for dimer-dimer collisions 3.33 for dimer-atom coll. (D. Petrov, C.S., G. Shlyapnikov) τ = 0.5 s τ = 20 ms Loss Rate: β ~ 2.4 10 -13 cm 3 /s First molecular BEC: 40 K 2 First molecular 40 2 JILA November 03 M. Greiner et al Nature Initial T=0.19 T F 4.7 10 5 molecules T=0.9 T c Initial T=0.06 T F 2 10 5 molecules T= 0.5 T C Molecule size: a/2= 1650 a 0 K 2 : a strongly interacting BEC 2 : a strongly interacting Condensed fraction Expansion energy per particle: mean field energy of molecular BEC is prop. to a Theory: a dd =0.6 a Condensation of Li 2 in Innsbruck create Fermi superfluid create Fermi superfluid Produce BEC of molecules with T/T BEC << 1 Cross the Feshbach resonance again adiabatically towards a < 0 a < 0 a > 0 According to L. Carr, G. Shlyapnikov and Y. Castin, cond-mat/0308306 this produces a deeply degenerate Fermi gas with T=0.01 T F or lower ! Thus T < T BCS for (k F a) final ~ 0.5 Unambiguous detection of superfluid required !! Perspectives Perspectives Strongly interacting Fermions BEC of molecules in the large a limit: non ideal effects Study crossover Condensate of molecules/ superfluid Fermi gas ( Nozières, Randeria , Ohashi, Griffin) Search for non ambiguous signature of fermionic superfluidity Mott transition for fermions in an optical lattice (Hoffstetter et al., 2002) Strongly correlated fermions domain is very vast ! Bosons/fermions mixtures ? Collapse of Fermi gas observed in Firenze ? Pairing mecanism mediated by bosons ? Improve on T/T F ? Better thermometry for T/T F <<1