Introduction to Cold Atoms and Bose-Einstein Condensation. Randy Hulet

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1 Introduction to Cold Atoms and Bose-Einstein Condensation Randy Hulet

2 Outline Introduction to methods and concepts of cold atom physics Interactions Feshbach resonances

3 Quantum Gases Quantum regime nλ 3 1 n = density Λ = de Broglie wavelength Identical particles! Gas phase n cm -3 Low temperature T 100 nk Λ 1 µm Phase transitions Bosons ( 7 Li): Bose-Einstein condensation Fermions ( 6 Li): Fermion pairing

4 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)

5 Methods Laser cooling T 100 µk Evaporative cooling T 100 µk 10 nk Atom trapping n cm -3, N trapping times ~ 1 minute Optical imaging

6 Imaging System

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9 Trapping Energy (MHz) F=3/2 0 F=1/ S 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 /2-1/2 1/ 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 ~ nm 532 nm

10 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

11 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)

12 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)

13 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

14 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

15 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

16 What Determines a? 7 Li 6 Li Zero-energy resonance a = -27 a o a = -2300? a o Answer: The last bound state!

17 Measuring a by Photoassociation 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

18 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

19 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

20 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/ /2-1/2 1/ 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)

21 Feshbach Resonances in 6 Li (Fermion) Scattering Length ( a O ) BEC bound states strongly interacting k F a 1 (attractive) Magnetic Field ( G ) Unitarity Limit BCS no bound states Houbiers, Stoof, McAlexander, Hulet, PRA 57, R1497 (1998) O Hara et al., PRA (2002)

22 Molecular BEC (using Fermions!) Normalized Column Density Normalized Column Density T < 0.1 T F N G k F a Length (μm) Thermal molecules plus BEC (partial evaporation) Length (μm) Pure BEC (full evaporation) PRL 95, (2005) Molecular BECs: JILA, Innsbruck, MIT, ENS, Duke

23 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) S 1/2 Ground State of 7 Li a < , Magnetic Field (G) 2,2 a (a 0 ) zero-crossing slope: 0.1 a o /G Magnetic Field (G)

24 Measuring a for a Bose-Einstein Condensate Optical trap ν z = 3 Hz / ν r = 192 Hz N º 3 ä Extract a by measuring the size of BEC vs. B R TF (Na) 1/5 resonance location R 4000 Axial Size (μm) zero crossing a z = h mω z 3000 a = 4.25 a resolution limit R Axial size is the 1/e radius Magnetic Field (G)

25 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 ) Thomas-Fermi Gaussian variation Including dipolar Using Thomas-Fermi or neglecting dipolar interactions produces systematic underestimate of the scattering length for small a Axial size (mm) e.g. Su Yi and Li You, PRA 67, (2003)

26 Scattering Length (a 0 ) Solitons Scattering Length vs. Field 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 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, (2009) 110um

27 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, (2006) (Stuttgart) Scattering Length (a 0 ) a Neglecting dipolar term 16 Hz axial trap b Including dipolar term Dipole interactions more important in skinny trap Hz axial trap Magnetic Field (G) Magnetic Field (G) Pollack et al., PRL 102, (2009)

28 Summary Making Quantum Gases laser cooling traps: magnetic and optical evaporative cooling Interactions two-body interactions s-wave scattering length tunable via Feshbach resonances

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