Introduction to Cold Atoms and Bose-Einstein Condensation Randy Hulet
Outline Introduction to methods and concepts of cold atom physics Interactions Feshbach resonances
Quantum Gases Quantum regime nλ 3 1 n = density Λ = de Broglie wavelength Identical particles! Gas phase n 10 12 cm -3 Low temperature T 100 nk Λ 1 µm Phase transitions Bosons ( 7 Li): Bose-Einstein condensation Fermions ( 6 Li): Fermion pairing
Lithium: Non-identical Twins 7 Li 6 Li 3 e s, 3 p s, 4 n s = 10 spin-½ particles Boson 94% abundance 3 e s, 3 p s, 3 n s = 9 spin-½ particles Fermion 6% abundance Truscott et al., Science 291, 2570 (2001)
Methods Laser cooling T 100 µk Evaporative cooling T 100 µk 10 nk Atom trapping n 10 12 10 14 cm -3, N 10 9 10 6 trapping times ~ 1 minute Optical imaging
Imaging System
Trapping Energy (MHz) 600 300 F=3/2 0 F=1/2-300 2S 1/2 State of 6 Li m F 3/2 1/2-1/2 m s = ½ Weak-field seeking magnetically trappable N N S S S N -600-3/2-1/2 1/2 0 100 200 300 400 500 Optical Traps 671 nm Field Strength (Gauss) F Δ = r I(r) Δ λ = 1060 nm m s = -½ Large Δ eliminates spontaneous emission Li Strong-field seeking not magnetically trappable 1060 nm w Aspect ratio ~ w/λ ~ 10-100 532 nm 532 nm
Evaporative Cooling trapped untrapped RF Spin flip hot atoms at edge of trap Collisions re-thermalize distribution colder denser f(e) Re-thermalize Remove tail E
Evaporative Cooling of Fermions Evaporative cooling requires rethermalizing collisions Pauli principle forbids s-wave interactions between identical fermions Ψ(1,2) = ψ spatial χ spin must be antisymmetric but ψ spatial = (-1) l = +1 for s-wave collisions Fermions 6 Li 6 Li Use both 6 Li and 7 Li 6 Li 7 Li 7 Li Bosons Other solutions: - Spin-state mixture in optical trap (JILA, Duke, Innsbruck) - BEC of sodium (MIT); BEC of lithium (ENS)
Bose-Einstein Condensation Rice (2002) Vortices Solitons Atom laser Atom wave guides/chips Nonlinear atom optics Collective oscillations Josephson oscillations Mott Insulator transition Disorder induced localization J.R. Abo-Shaeer, MIT (2001)
Interest in Ultracold Fermi Gases Strongly interacting fermions: high-t c superconductors neutron stars quark-gluon plasma Connections to condensed matter Hubbard model of high-t c Pseudo-gap physics Strong correlations in low dimensions: Luttinger liquid, spin-charge separation, Optical lattice configurations 1D 2D 3D
Interactions Generic Discussion V(R) 2-body potential R Λ db ~ 1 μm detailed shape of V(R) unimportant ~1 nm Characterize interaction by s-wave scattering length a: mean-field interaction energy nu o = 4πh 2 na/m a < 0 attractive a > 0 repulsive
Bosons Implications of Interactions stability of condensate bright or dark solitons healing length: vortices, speed of sound excitation spectrum Mott insulator: on-site interactions U miscibility or immiscibility of spinor condensates Fermions Cooper pairing (BCS) or molecules (BEC) Hubbard model: U/J
What Determines a? 7 Li 6 Li Zero-energy resonance a = -27 a o a = -2300? a o Answer: The last bound state!
Measuring a by Photoassociation + 1 3 Σ g 7 Li 2 v' Energy a 3 Σ u + ω 2 ω 1 v = 10 Interatomic Separation Abraham et al., PRL 74, 1315 (1995) 2-photon photoassociation: E B = ω 2 - ω 1 Li potentials well characterized: 2-body physics known precisely
Tuning Interactions: BEC-BCS Crossover w Fermions BCS pairing crosses over to Bose-Einstein condensation of molecules with increasing U o : BCS Unitarity k F a BEC U o R Interactions determined by the s-wave scattering length a: g 2 4π h a = m a U o
Feshbach Resonance Magnetically tune free atoms into resonance with a bound molecular state: electronic spin detuning δ - ΔμB (S = 0) closed channel + open channel (S = 1) interatomic separation
Feshbach Resonance in 6 Li (Fermions) 600 2S 1/2 State of 6 Li m F 3/2 1/2-1/2 Energy (MHz) 300 F=3/2 0 F=1/2-300 -600-3/2-1/2 1/2 0 100 200 300 400 500 Field Strength (Gauss) s-wave pairing but in an electronic triplet state Quasi-spin ½: spin-down 2 spin-up 1 Energy atoms molecule molecule μ = 0 (S = 0) Magnetically tuned collisional resonance B atoms μ = -2μ B (S = 1)
Feshbach Resonances in 6 Li (Fermion) Scattering Length ( a O ) 10000 5000 0-5000 BEC bound states strongly interacting k F a 1 (attractive) -10000 400 600 800 1000 1200 Magnetic Field ( G ) Unitarity Limit BCS no bound states Houbiers, Stoof, McAlexander, Hulet, PRA 57, R1497 (1998) O Hara et al., PRA 66 041401 (2002)
Molecular BEC (using Fermions!) Normalized Column Density 1.0 0.5 0.0 Normalized Column Density 1.0 0.5 0.0 T < 0.1 T F N 7 10 4 695 G k F a 0.14-500 0 500 Length (μm) Thermal molecules plus BEC (partial evaporation) -500 0 500 Length (μm) Pure BEC (full evaporation) PRL 95, 020404 (2005) Molecular BECs: JILA, Innsbruck, MIT, ENS, Duke
Feshbach Resonance in 7 Li (Boson) Hyperfine sublevels of 7 Li Coupled channels calculation of the scattering length of 7 Li 1,1 state Energy (MHz) 1800 1350 2S 1/2 Ground State of 7 Li a < 0 900 450 0-450 -900-1350 1,1-1800 0 200 400 600 800 1000 Magnetic Field (G) 2,2 a (a 0 ) 100 50 0-50 -100 zero-crossing slope: 0.1 a o /G 0 200 400 600 800 1000 Magnetic Field (G)
Measuring a for a Bose-Einstein Condensate Optical trap ν z = 3 Hz / ν r = 192 Hz N º 3 ä 10 5 1000 Extract a by measuring the size of BEC vs. B R TF (Na) 1/5 resonance location R 4000 Axial Size (μm) 100 10 zero crossing a z = h mω z 3000 a = 4.25 a 0 2000 resolution limit 1000 0 R -0.4-0.2 0.0 0.2 0.4 Axial size is the 1/e radius 1 550 600 650 700 750 Magnetic Field (G)
Extracting the Scattering Length Solve GP equation (including magnetic dipolar interactions) by minimizing energy with Gaussian wavefunction: Compare measured 1/e axial radius to calculated Gaussian radius to extract a 1000 Scattering length (a o ) 100 10 1 0.1 0.01 Thomas-Fermi Gaussian variation Including dipolar Using Thomas-Fermi or neglecting dipolar interactions produces systematic underestimate of the scattering length for small a 0.01 0.1 Axial size (mm) e.g. Su Yi and Li You, PRA 67, 045601 (2003)
Scattering Length (a 0 ) Solitons 10 6 10 4 10 2 10 0 10-2 Scattering Length vs. Field 0.4 0.2 0.0-0.2 540 542 544 546 548 a = 24.5a ( 0 Quasi 1D regime Coupled channels calculation 192.3G ) Derived 1+ scattering length B 736.9G Coupled channels calculation Feschbach resonance fit 550 600 650 700 B=543.36G Magnetic Field (G) e.g. Khaykovich et al., Science 296, 1290 (2002) (ENS) and Strecker et al., Nature 417, 150 (2002) (Rice) 3D regime Extremely shallow slope: 0.1 a o /G B=736.8G 1400um na 3 à 1 >7 decades Pollack et al., PRL 102, 090402 (2009) 110um
Effect of Dipolar Interactions Magnetic dipole-dipole interactions important when: a < 2 μ o μ m m 2 12π h = 0.6 a o for 7 Li Griesmaier et al., PRL 97, 250402 (2006) (Stuttgart) Scattering Length (a 0 ) 0.3 0.2 0.1 a Neglecting dipolar term 16 Hz axial trap b Including dipolar term Dipole interactions more important in skinny trap 0.0 3 Hz axial trap 544 545 546 544 545 546 Magnetic Field (G) Magnetic Field (G) Pollack et al., PRL 102, 090402 (2009)
Summary Making Quantum Gases laser cooling traps: magnetic and optical evaporative cooling Interactions two-body interactions s-wave scattering length tunable via Feshbach resonances