Superfluid Bose and Fermi gases Wolfgang Ketterle Massachusetts Institute of Technology MIT-Harvard Center for Ultracold Atoms 3/11/2013 Universal Themes of Bose-Einstein Condensation Leiden Superfluidity and Superconductivity Macroscopic Quantum Phenomena 1
Single particle quantum mechanics Single particle quantum mechanics Many particle quantum mechanics 2
Single particle quantum mechanics Many particle quantum mechanics Macroscopic quantum phenomena Single particle quantum mechanics Many particle quantum mechanics Macroscopic quantum phenomena 3
This phenomenon, called Bose-Einstein condensation, is at the heart of superfluidity and superconductivity * 1925 4
Why did it take 70 years to realize BEC in a gas? Criterion for BEC Thermal de Broglie wavelength ( T -1/2 ) equals distance between atoms (= n -1/3 ) n crit T 3/2 High density: n water : T = 1 K BUT: molecule/cluster formation, solidification no BEC Low density: n water /10 9 : T= 100 nk - 1 µk seconds to minutes lifetime of the atomic gas BEC 5
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Hydrogen advantage: Three body recombination rate coefficient is ten orders of magnitude smaller but: elastic cross section much smaller Sodium BEC window for alkalis is larger than for hydrogen (and at lower density) 7
BEC @ JILA, June 95 (Rubidium) BEC @ MIT, Sept. 95 (Sodium) 8
Rotating superfluids Superfluid described by macroscopic matter wave A superfluid is irrotational unless Velocity field: Vorticity can enter the superfluid only in singularities, the vortices When going around a closed loop, φ can only change by multiples of 2π 9
Rotating condensates non-rotating rotating (160 vortices) J. Abo-Shaeer, C. Raman, J.M. Vogels, W.Ketterle, Science, 4/20/2001 Two-component vortex Boulder, 1999 Single-component vortices Paris, 1999 Boulder, 2000 MIT 2001 Oxford 2001 10
Discovery of superfluidity and superconductivity 1908 Liquefaction of helium (Kamerlingh-Onnes): Superfluid helium created, but not recognized 1911 Discovery of superconductivity in mercury (Kamerlingh-Onnes) >100 years 1937 Discovery of superfluidity in helium (Kapitza, Allen, Misener) 1911: First fermionic superfluids (superconducting mercury) were sympathetically cooled by ultracold bosons (liquid helium) Recently: Fermi gases (e.g. 6 Li) are cooled by sympathetic cooling (evaporative cooling of bosonic gases, e.g. Na) 1911/1938: Transition temperatures of He-II (2.2 K) and mercury (4.2 K), tin (3.8 K), lead (6 K) similar: purely technical reasons 11
Degeneracy temperature same for bosons and fermions (at the same density and mass) In cold gases: typically 200 nk 2 µk Footnote: in extreme cases 0.5 5 nk To avoid inelastic collisions ( 85 Rb, JILA) Low density for atom interferometry ( 87 Rb, Virginia) To achieve normal-incidence quantum reflection (Na, MIT) Superfluidity in fermions: Usually requires much lower temperatures than degeneracy temperature Kamerlingh-Onnes: exponential ( pairing ) factor was equal to T fermi (electrons)/t degeneracy ( 4 He) But: Exponential factor is unity for a 12
1911: First fermionic superfluids (superconducting mercury) were sympathetically cooled by ultracold bosons (liquid helium) Recently: Fermi gases (e.g. 6 Li) are cooled by sympathetic cooling (evaporative cooling of bosonic gases, e.g. Na) 1911/1938: Transition temperatures of He-II (2.2 K) and mercury (4.2 K), tin (3.8 K), lead (6 K) similar: purely technical reasons Recently: Transition temperatures of Bose and Fermi gases similar (Fermi gas with unitarity limited interactions): fundamental (?) unitarity limit Transition temperature Fermi temperature (density) 2/3 10-5 10-4 normal superconductors 10-3 superfluid 3 He 10-2 high T c superconductors 0.15 high T c superfluid 13
How to vary a? How to get a? Pair A-B Particle A Particle B 14
Resonant interactions have infinite strength Pair A-B Particle A Particle B E Free atoms Molecule Magnetic field Feshbach resonance 15
Disclaimer: Drawing is schematic and does not distinguish nuclear and electron spin. E Free atoms Molecule Magnetic field Feshbach resonance form a stable molecule E Free atoms Molecule Magnetic field Feshbach resonance 16
form an unstable molecule E Free atoms Molecule Magnetic field Feshbach resonance Atoms attract each other E Free atoms Molecule Magnetic field Feshbach resonance 17
Atoms repel each other Atoms attract each other E Free atoms Molecule Magnetic field Feshbach resonance Atoms repel each other Atoms attract each other Force between atoms Scattering length Magnetic field Feshbach resonance 18
Energy Feshbach Resonance Atoms Molecules Magnetic field Atoms repel each other Atoms attract each other Force between atoms Scattering length Magnetic field Feshbach resonance 19
Feshbach Resonance Energy Atoms Molecules Magnetic field Atoms form stable molecules Atoms repel each other a>0 BEC of Molecules: Condensation of tightly bound fermion pairs Molecules are unstable Atoms attract each other a<0 BCS-limit: Condensation of long-range Cooper pairs Atom pairs Electron pairs Bose Einstein condensate of molecules BCS Superconductor 20
BEC BCS supe Magnetic field BEC BCS supe 21
BEC Crossover superfluid BCS supe Preparation of an interacting Fermi system in 6 Li Setup: Optical trapping: 9 W @ 1064 nm ω = 2π (16,16, 0.19) khz E trap = 800 µk States 1> and 2> correspond to > and > 22
Cold atomic gases: Realization of an s-wave fermionic superfluid in the strong coupling limit of BCS theory Evidence for phase transition Bose-Einstein condensation Peaks in the momentum distribution (visible in spatial distribution after ballistic expansion) Superfluidity Vortex lattices for rotating gas JILA, Nature 426, 537 (2003). MIT, PRL 91, 250401 (2003) Innsbruck, PRL 92, 120401 (2004). ENS, PRL 93, 050401 (2004). 23
Vortex lattices in the BEC-BCS crossover M.W. Zwierlein, J.R. Abo-Shaeer, A. Schirotzek, C.H. Schunck, W. Ketterle, Nature 435, 1047-1051 (2005) Superfluidity of fermions requires pairing of fermions Microscopic study of the pairs by RF spectroscopy 24
RF spectroscopy 3> > hf 0 3> > hf 0 + > > Dissociation spectrum measures the Fourier transform of the pair wavefunction Width (1/pair size) 2 Threshold (1/pair size) 2 25
Rf spectra in the crossover Standard superconductors ξ>> 1/k F High T c superconductors ξ 6 10 (1/k F ) Superfluid at unitarity ξ = 2.6 (1/k F ) Interparticle spacing ~ 3.1 (1/k F ) Molecular character of fermion pairs Confirms correlation between high T c and small pairs C. H. Schunck, Y. Shin, A. Schirotzek, W. Ketterle, Nature 454, 739 (2008). Benchmarking the Fermi gas at unitarity 26
Equation of State: Measuring density Exploiting cylindrical symmetry and careful characterization of trapping potential: Density [10 11 /cm 3 ] 2.5 2.0 1.5 1.0 0.5 0.0 0.0 Experimental n(v) from single profile 0.4 0.8 V [µk] 1.2 For a resonant gas: 2E P = 3 V Heat capacity ( ENkB ) T P κ0 C d / 3 V F = =... = NkB dt 2 T P0 κ NV, k B 1.5 Specific Heat C V /N 1.0 0.5 0.0 0.0 0.5 1.0 Unitary Fermi Gas Direct observation of the superfluid transition at T C /T F = 0.167(13) 1.5 T/T F Mark J. H. Ku, Ariel T. Sommer, Lawrence W. Cheuk, Martin W. Zwierlein Science 335, 563-567 (2012) 27
Experimental realization of the BCS-BEC crossover Theory: Late 60s: Popov, Keldysh, Eagles 80 s Leggett, Nozières, Schmitt-Rink Demonstrates that BEC and BCS are two limiting cases of one theory BCS-BEC Crossover Superfluid transition temperature BCS Strength of interactions Highest fermionic transition temperature BEC 28
The BEC-BCS Crossover The BCS wave function Φ + + = ( uk vkc c ) vac BCS + k k k can be written as a BEC wave function of pairs b + + + = ϕkc c k k k + Φ = ( b ) N vac BEC This was known already soon after BCS theory was formulated. F. However, Dyson (1957, the pair cited creation by Bardeeen) and annihilation operators fulfill bosonic commutation relationships only in the BEC limit of small pair size Now generally accepted: Superconductivity is kind of a Bose-Einstein condensate of electron pairs. However: + Φ = ( b ) N vac BEC Overlapping electron pairs are modified by Pauli exclusion principle 29
Novelty of ultracold atomic gases: Many body physics is realized in an ultra-dilute gas, at densities a billion times less than solids and liquids Superfluidity in a gas Realization of systems with truly short-range interactions What are the simplest interactions? Short range shorter than any other length scale interatomic distance, de Broglie wavelength characterized by only one parameter (strength) approximated by delta functions momentum space scattering length 30
Ultracold collisions At ultralow temperatures, only s-wave ( head-on ) collisions remain Collisions parametrized by one single quantity: scattering length a de Broglie wavelength >> range of interatomic potential R ~50 a 0 ~2 nm Ultracold collisions At ultralow temperatures, only s-wave ( head-on ) collisions remain Collisions parametrized by one single quantity: scattering length a de Broglie wavelength >> range of interatomic potential r R λ db ~ µm 31
Quantum simulators New materials harnessing strong correlations in many-electron systems: Nanotubes, quantum magnets, superconductors, Quantum simulators: Controlled,? Approximations, Impurities, simple systems testing models and verifying concepts no exact solutions Condensed matter models: Simple models which capture the relevant mechanism Cold atomic gases provide the building blocks of quantum simulators Quantum engineering of interesting Hamiltonians Ultracold Bose gases: superfluidity (like 4 He) Ultracold Fermi gases (with strong interactions near the unitartiy limit): pairing and superfluidity (BCS, like superconductors) Now: strongly correlated systems 32
New frontiers: Interactions at the unitarity limit Synthetic gauge field Rapidly rotating gases Quantum Hall effect Spin-orbit coupling Disorder Anderson localization Few-body correlations, Effimov states Long-range interactions (Rydberg, dipolar) 33
BEC II Ultracold fermions 6 Li: Lattice density fluct. Ed Su Wujie Huang Junru Li (Aviv Keshet) (C. Sanner) (J. Gillen) $$ NSF ONR ARO MURI-AFOSR MURI-ARO DARPA BEC III Na-Li molecules Repulsive fermions Tout Wang Timur Rvachov Chenchen Luo Myoung-Sun Heo (Dylan Cotta) (Ye-Ryoung Lee) BEC IV Rb BEC in optical lattices Hiro Miyake Georgios Siviloglou Colin Kennedy Cody Burton (David Weld (UCSB)) D.E. Pritchard BEC V New exp: 7 Li in optical lattices Jesse Amato-Grill Ivana Dimitrova Niklas Jepsen (David Weld (UCSB)) (Graciana Puentes) 34