Introduction to Ultracold Atomic Gases

1. Introduction to Ultracold Atomic Gases Qijin Chen

2. What is an ultracold atomic gas? • Gases of alkli atoms, etc • How cold is cold? • Microwave background 2.7K • He-3: 1 mK • Cornell and Wieman 1 nK • Quantum degeneracy • Bose/Fermi/Boltzmann Statistics

3. Bose/Fermi/Boltzmann Statistics • Boltzmann • Bose • Fermi

4. Chemical potential  • Particle number constraint • Boltzmann •  < 0 • Bose • Fermi

5. Quantum degenerate particles: fermions vs bosons T = 0 EF = kBTF spin spin Bose-Einstein condensation Fermi sea of atoms Pauli exclusion

6. Quantum degeneracy condition • Ultracold Fermi gases • or lower Bose gases

7. Laser cooling – Brief history • Cooling atoms to get better atomic clocks • In 1978, researchers cooled ions somewhat below 40 Kelvin; ten years later, neutral atoms had gotten a million times colder, to 43 microkelvin. • Basic physics: use the force of laser light applied to atoms to slow them down. • Higher K.E. + lower photon energy = lower K.E. + higher photon energy • In 1978 Dave Winelan @ NIST, CO – Laser cooled ions using Doppler cooling techniche. Laser tuned just below the resonance frequency. • In 1982, William Phillips (MIT -> NIST@Gaithersburg, MD) and Harold Metcalf (Stony Brook University of NY) laser cooled neutral atoms

8. Laser cooling (cont’d) • Late 1980s – 240 K for Na, thought to be the lowest possible – Doppler limit. • In 1988, – 43 K. A Phillips’ group accidentally discovered that a technique developed three years earlier by Steven Chu and colleagues at Bell Labs in New Jersey [3] could shatter the Doppler limit. • Later in 1988, Claude Cohen-Tannoudji of the École Normale Supérieure in Paris and his colleagues broke the "recoil" limit [4]--another assumed lower limit on cooling. • In1995, creation of a Bose-Einstein condensate • 1997 Nobel Prize in physics • Details @ http://focus.aps.org/story/v21/st11

10. BEC in Bosonic Alkali Atoms • BEC – Fifth state of matter • BEC was predicted in 1924 • Achieved in dilute gases of alkali atoms in 1995 • Nobel Prize in physics 2001: 50 nK 200 nK 400 nK Eric A Cornell, Carl E Wieman, Wolfgang Ketterle 87Rb

11. Momentum distribution

12. Momentum distribution of a BEC 50 nK 200 nK 400nK http://www.colorado.edu/physics/2000/bec/index.html

13. Rubidium-78 Cornell and Wieman

14. Na-23: Ketterle @ MIT • 4 month later, 100 times more atoms

15. Density distribution of a condensate • Simple harmonic oscillator Fourier transform of Gaussian is also Gaussian

16. Phase coherence – Interference pattern – Ketterle @ MIT

17. Physics of BEC – Bose Statistics

18. Gross-Pitaeviskii Equation • Interacting, inhomogeneous Bose gases • Condensate wavefunction • Condensate density • total number of atoms • Total energy: • Minimizing energy: V(r) – External potential, U0 -- Interaction

19. Atomic Fermi gases • Moved on to cooling Fermi atoms • Achieved Fermi degeneracy in 1999 by Debbie Jin • Molecular condensate achieved in 2003 by Jin, Ketterle and Rudi Grimm @ Innsbruck, Austria. • Jin quickly created the first Fermi condensate, composed of Cooper pairs.

20. Superfluidity in Fermi Systems • Discovery of superconductivity, 1911 • Heike Kamerlingh Onnes • Nobel Prize -1913 • Theory of superconductivity, 1957 • J. Bardeen, L.N. Cooper, J.R. Schrieffer • Nobel Prize -1972 • High Tc superconductors, 1986 • J.G. Bednorz, K.A. Müller • Nobel Prize - 1987 • Discovery of superfluid 3He, 1972 • D.M. Lee, D.D. Osheroff R.C. Richardson • Nobel Prize - 1996 • Nobel Prize in physics 2003 • A.A. Abrikosov (vortex lattice) • V.L. Ginzburg (LG theory) • A.J. Leggett (superfluid 3He)

21. Where will the next Nobel Prize be?

22. Superfluidity in Atomic Fermi Gases • Quantum degenerate atomic Fermi gas – 1999 • B. DeMarco and D. S. Jin, Science 285, 1703 (1999) • Creation of bound di-atomic molecules – 2003 • 40K: Jin group (JILA), Nature 424, 47 (2003). • 6Li: Hulet group (Rice),PRL 91, 080406 (2003); • 6Li: Grimm group (Innsbruck), PRL 91, 240402 (2003) • Molecular BEC from atomic Fermi gases – Nov 2003 • 40K: Jin group, Nature 426, 537 (2003). • 6Li: Grimm group, Science 302, 2101 (2003) • 6Li: Ketterle group (MIT), PRL 91, 250401 (2003). • Fermionic superfluidity (Cooper pairs) – 2004 • Jin group, PRL 92, 040403 (2004) • Grimm group, Science 305, 1128 (2004) • Ketterle group, PRL 92, 120403 (2004).

23. Superfluidity in Atomic Fermi Gases • Molecular BEC from atomic Fermi gases – Nov 2003 • 40K: Jin group, Nature 426, 537 (2003). • 6Li: Grimm group, Science 302, 2101 (2003) • 6Li: Ketterle group (MIT), PRL 91, 250401 (2003). • Fermionic superfluidity (Cooper pairs) – 2004 • Jin group, PRL 92, 040403 (2004) • Grimm group, Science 305, 1128 (2004) • Ketterle group, PRL 92, 120403 (2004). • Heat capacity measurement + thermometry in strongly interacting regime – 2004 • Thomas group (Duke) + Levin group (Q. Chenet al., Chicago), Science Express, doi:10.1126/science.1109220 (Jan 27, 2005)

24. What are Cooper pairs? Cooper pair is the name given to electrons that are bound together at low temperatures in a certain manner first described in 1956 by Leon Cooper.[1] Cooper showed that an arbitrarily small attraction between electrons in a metal can cause a paired state of electrons to have a lower energy than the Fermi energy, which implies that the pair is bound.

25. Where does the attractive interaction come from? • In conventional superconductors, electron-phonon (lattice) interaction leads to an attractive interaction between electrons near Fermi level. • An electron attracts positive ions and draw them closer. When it leaves, also leaving a positive charge background, which then attracts other electrons.

26. Feshbach resonances in atoms • Atoms have spins • Different overall spin states have different scattering potential between atoms • -- different channels • Open channel – scattering state • Closed channel – two-body bound or molecular state

27. → ← R R R R DB > molecules Tuning interaction in atoms via a Feshbach resonance V(R) a>0, strong attraction bound state a<0, weak attraction We can control attraction via B field !

28. Tuning interaction via a Feshbach resonance 6Li

29. Introduction to BCS theory • 2nd quantization – quantum field theory – many-body theory • Fermi gases

30. Interactions • Interaction energy Neglecting the spin indices

31. Reduced BCS Hamiltonian • Only keep q=0 terms of the interaction • Bogoliubov transformation

32. = Order Parameter Self-consistency condition leads to gap equation

33. Overview of BCS theory Fermi Gas BCS superconductor No excitation gap

34. BCS theory works very well for weak coupling superconductors

35. Facts About Trapped Fermi Gases • Mainly 40K (Jin, JILA;Inguscio, LENS) and 6Li (Hulet, Rice; Salomon, ENS; Thomas, Duke; Ketterle, MIT; Grimm, Innsbruck) • Confined in magnetic and optical traps • Atomic number N=105-106 • Fermi temperature EF ~ 1 mK • Cooled down to T~10-100 nK • Two spin mixtures – (pseudo spin) up and down • Interaction tunable via Feshbach resonances

36. EF spin  spin  Making superfluid condensate with fermions • BEC of diatomic molecules • BCS superconductivity/superfluidity • 1. Bind fermions together. • BEC • Attractive interaction needed Condensation of Cooper pairs of atoms (pairing in momentum space)

37. Lecture 2

38. BCS PG/Unitary BEC Physical Picture of BCS-BEC crossover:Tuning the attractive interaction • Change of character: fermionic ! Bosonic • Pairs form at high T (Uc – critical coupling) •   DSC , TC T* Exists a pseudogap • Two types of excitations

39. High Tc superconductors: Tuning parameter: hole doping concentration Increasing interaction Cannot reach bosonic regime due to d-wave pairing

40. Crossover and pseudogap physics in high Tc superconductors • BCS-BEC crossover provides a natural explanation for the PG phenomena. Q. Chen, I. Kosztin, B. Janko, and K. Levin, PRL 81, 4708 (1998) BSCCO, H. Ding et al, Nature 1996

41. Molecules of fermionic atoms Cooper pairs kF BEC of bound molecules BCS superconductivity Cooper pairs: correlated momentum-space pairing Crossover under control in cold Fermi atoms (1st time possible) Magnetic Field hybridized Cooper pairs and molecules Pseudogap / unitary regime Theoretical study of BCS-BEC crossover:Eagles, Leggett, Nozieres and Schmitt-Rink, TD Lee, Randeria, Levin, Micnas, Tremblay, Strinati, Zwerger, Holland, Timmermans, Griffin, … Attraction

42. Terminology • Molecules – Feshbach resonance induced molecular bosons – Feshbach molecules – Feshbach bosons --- Should be distinguished from Cooper pairs -- many-body effect induced giant pairs • Unitarity – unitary limit -- where a diverges This is the strongly interacting or pseudogap phase (  DSC , TC T* ) • BEC limit : • Strong attractive interaction – fermions • Weak repulsive interaction – bosons or pairs

43. Big questions – • Cold atoms may help understanding high Tc • How to determine whether the system is in the superfluid phase? • Charge neutral • Existence of pseudogap • How to measure the temperature? • Most interesting is the pseudogap/unitary regime – diverging scattering length – strongly interacting

44. Evidence for superfluidity Ti/TF = 0.19 0.06 Time of flightabsorption image Molecular CondensateBimodal density distributionAdiabatic/slow sweep from BCS side to BEC side. Molecules form and Bose condense. M. Greiner, C.A. Regal, and D.S. Jin, Nature 426, 537 (2003).

45. Cooper pair condensate Dissociation of molecules at low density C. Regal, M. Greiner, and D. S. Jin, PRL 92, 040403 (2004) DB (gauss) DB = 0.12 GDB = 0.25 GDB=0.55 G T/TF=0.08

46. Observation of pseudogap-Pairing gap measurements using RF - C. Chin et al, Science 305, 1128 (2004) Torma’s theoretical calculation based on our theory

47. Highlights of previous work on high Tc Extended ground state crossover to finite T, with a self- consistent treatment of the pseudogap. • Phase diagram for high Tc superconductors, in (semi-) quantitative agreement with experiment. • Quasi-universal behavior of superfluid density. • The only one in high Tc that is capable of quantitative calculations • We are now in a position to work on cold atoms Q. Chen et al, PRL 81, 4708 (1998)

48. Highlights of our work on cold atoms • The first one that introduced the pseudogap to cold atom physics, calculated Tc, superfluid density, etc • Signatures of superfluidity and understanding density profiles PRL 94, 060401 (2005)

49. Highlights of our work on cold atoms • First evidence (with experiment) for a superfluid phase transition Science Express, doi:10.1126/science.1109220 (2005) • Thermodynamic properties of strongly interacting trapped gases

50. Summary • Ultracold Fermi gases near Feshbach resonances are a perfect testing ground for a crossover theory due to tunable interactions. • Will help understanding high Tc problem. • Signature of superfluidity in the crossover / unitary regime is highly nontrivial. • We and Duke group have found the strongest evidence for fermionic superfluidity. • In the process, we developed thermometry.