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Introduction to Ultracold Atomic Gases Qijin Chen 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 Bose/Fermi/Boltzmann Statistics • Boltzmann • Bose • Fermi Chemical potential • Particle number constraint • Boltzmann – <0 • Bose • Fermi Quantum degenerate particles: fermions vs bosons EF = kBTF T=0 spin spin Bose-Einstein condensation Fermi sea of atoms Pauli exclusion Quantum degeneracy condition • Ultracold Fermi gases • or lower Bose gases is the critical temperature, is the particle density, is the mass per boson, is Reduced Planck's constant, is the Boltzmann constant, and is the Riemann zeta function; 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 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 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 -- 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 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 Momentum distribution Momentum distribution of a BEC 50 nK 200 nK 400 nK http://www.colorado.edu/physics/2000/bec/index.html Rubidium-78 Cornell and Wieman Na-23: Ketterle @ MIT • 4 month later, 100 times more atoms Density distribution of a condensate • Simple harmonic oscillator Fourier transform of Gaussian is also Gaussian Phase coherence – Interference pattern – Ketterle @ MIT Physics of BEC – Bose Statistics 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 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. 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 3 Discovery of superfluid He, 1972 J.G. Bednorz, K.A. Müller D.M. D.D. Osheroff NobelLee, Prize - 1987 R.C. Richardson Nobel Prize in physics 2003 Nobel Prize - 1996 A.A. Abrikosov (vortex lattice) V.L. Ginzburg (LG theory) A.J. Leggett (superfluid 3He) Superfluids – A Nobel Prize producing field Superfluidity in Atomic Fermi Gases Quantum degenerate atomic Fermi gas – 1999 Creation of bound di-atomic molecules – 2003 40K: 6Li: Jin group (JILA), Nature 424, 47 (2003). Hulet group (Rice), PRL 91, 080406 (2003); 6Li: Grimm group (Innsbruck), PRL 91, 240402 (2003) Molecular BEC from atomic Fermi gases – Nov 2003 B. DeMarco and D. S. Jin, Science 285, 1703 (1999) 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). 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. Chen et al., Chicago), Science Express, doi:10.1126/science.1109220 (Jan 27, 2005) 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. Where does the attractive interaction come from? • In conventional superconductors, electronphonon (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. 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 Tuning interaction in atoms via a Feshbach resonance V(R) R a>0, strong attraction bound state molecules R R R a<0, weak attraction →← > DB We can control attraction via B field ! Tuning interaction via a Feshbach resonance 0 6Li Introduction to BCS theory 2nd quantization – quantum field theory – manybody theory Fermi gases Interactions • Interaction energy Neglecting the spin indices Reduced BCS Hamiltonian • Only keep q=0 terms of the interaction • Bogoliubov transformation D = Order Parameter Self-consistency condition leads to gap equation Overview of BCS theory Fermi Gas No excitation gap BCS superconductor BCS theory works very well for weak coupling superconductors 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 K Cooled down to T~10-100 nK Two spin mixtures – (pseudo spin) up and down Interaction tunable via Feshbach resonances Making superfluid condensate with fermions BEC of diatomic molecules 1. Bind fermions together. 2. BEC 3. Attractive interaction needed spin spin BCS superconductivity/superfluidity Condensation of Cooper pairs of atoms (pairing in momentum space) EF Lecture 2 Physical Picture of BCS-BEC crossover: Tuning the attractive interaction Change of character: fermionic ! Bosonic Pairs form at high T (Uc – critical coupling) D DSC , TC T* Exists a pseudogap Two types of excitations BCS PG/Unitary BEC High Tc superconductors: Tuning parameter: hole doping concentration Increasing interaction Cannot reach bosonic regime due to d-wave pairing 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 Crossover under control in cold Fermi atoms (1st time possible) Magnetic Field Molecules of fermionic atoms hybridized Cooper pairs and molecules Cooper pairs kF BEC of bound molecules Pseudogap / unitary regime BCS superconductivity Cooper pairs: correlated momentum-space pairing Theoretical study of BCS-BEC crossover: Eagles, Leggett, Nozieres and Schmitt-Rink, TD Lee, Randeria, Levin, Micnas, Tremblay, Strinati, Zwerger, Holland, Timmermans, Griffin, … Attraction 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 Evidence for superfluidity Time of flight absorption image Ti/TF = 0.19 0.06 Molecular Condensate Bimodal density distribution Adiabatic/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). Cooper pair condensate N molecules 5 3x10 5 Dissociation of molecules at low density 2x10 5 1x10 0 -0.5 0.0 DB (gauss) DB = 0.12 G DB = 0.25 G T/TF=0.08 0.5 C. Regal, M. Greiner, and D. S. Jin, PRL 92, 040403 (2004) DB=0.55 G Observation of pseudogap -Pairing gap measurements using RF C. Chin et al, Science 305, 1128 (2004) Torma’s theoretical calculation based on our theory Highlights of application on high Tc Extended ground state crossover to finite T, with a selfconsistent 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 Q. Chen et al, PRL 81, 4708 (1998) Highlights of application 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) 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 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. Theoretical Formalism and Results Grand canonical Hamiltonian for resonance superfluidity Our solution has the following features: 1. BCS-like ground state: 2. Treat 2-particle and 1-particle propagators on an equal footing – including finite momentum (bosonic) pair excitations self-consistently. T-matrix formalism Integrate out boson field: T-matrix t(Q)= Fermion self-energy: D2 = Dpg2 + Dsc2 Self-consistent Equations Gap equation: BEC condition Number equation: Chemical potential Pseudogap equation: Pair density Critical temperature Homogeneous case: • Maximum at resonance, minimum at =0 • BCS at high field, BEC at low field In the trap: • Local density approximation: ! - V(r) • Tc increases with decreasing 0 due to increasing n(r=0) Understanding the profiles at unitarity PRL 94, 060401 (2005) • Theoretical support to TF based thermometry in the strongly interacting regime Uncondensed pairs smooth out the profiles Profile decomposition Condensate Noncondensed pairs Fermions Thermodynamics of Fermi gases Bosonic contribution to thermodynamic potential Entropy: fermionic and bosonic. Entropy of Fermi gases in a trap Power law different from noninteracting Fermi or Bose gases Fall in between, power law exponent varies. Can be used to determine T for adiabatic field sweep experiments Thermodynamics of Fermi gases Science Express, doi:10.1126/science.1109220 (2005) Experimental T Temperature calibrated to account for imperfection of TF fits • Very good quantitative agreement with experiment Conclusions Interaction between ultracold fermions can be tuned continuously from BCS to BEC. This may eventually shed light on high Tc superconductivity. Except in the BCS regime, opening of an excitation gap can no longer be taken as a signature of superfluidity. Pseudogap makes these gases more complicated and interesting. Our theory works very well in fermionic superfluidity in cold atoms. A whole new field Interface of AMO and condensed matter physics Cooper pairs of electrons in superconductors Excitons in semiconductors 3He Alkali atoms in ultracold atom gases Neutron pairs, proton pairs in nuclei And neutron stars atom pairs in superfluid 3He-A,B Mesons in neutron star matter