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

Boltzmann Bose Fermi

Chemical potential  Particle number constraint Boltzmann –  < 0 Bose Fermi

Quantum degenerate particles: fermions vs bosons Bose-Einstein condensation Fermi sea of atoms E F = k B T F spin  spin  T = 0 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 constantReduced Planck's constant, is the Boltzmann constant, andBoltzmann constant is the Riemann zeta function; 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 NIST, CO – Laser cooled ions using Doppler cooling techniche. Laser tuned just below the resonance frequency. In 1982, William Phillips (MIT -> 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 in New Jersey [3] could shatter the Doppler limit.3 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.4 In1995, creation of a Bose-Einstein condensate 1997 Nobel Prize in physics

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

Momentum distribution

Momentum distribution of a BEC 400 nK 200 nK 50 nK

Rubidium-78 Cornell and Wieman

Na-23: 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 – 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, U 0 -- 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 Innsbruck, Austria. Jin quickly created the first Fermi condensate, composed of Cooper pairs.

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

Where will the next Nobel Prize be?

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

Superfluidity in Atomic Fermi Gases Molecular BEC from atomic Fermi gases – Nov 2003 Molecular BEC from atomic Fermi gases – Nov K: Jin group, Nature 426, 537 (2003). 40 K: Jin group, Nature 426, 537 (2003). 6 Li: Grimm group, Science 302, 2101 (2003) 6 Li: Grimm group, Science 302, 2101 (2003) 6 Li: Ketterle group (MIT), PRL 91, (2003). 6 Li: Ketterle group (MIT), PRL 91, (2003). Fermionic superfluidity Fermionic superfluidity (Cooper pairs) – 2004 Jin group, PRL 92, (2004) Jin group, PRL 92, (2004) Grimm group, Science 305, 1128 (2004) Grimm group, Science 305, 1128 (2004) Ketterle group, PRL 92, (2004). Ketterle group, PRL 92, (2004). Heat capacity measurement + thermometry in strongly interacting regime – 2004 Heat capacity measurement + thermometry in strongly interacting regime – 2004 Thomas group (Duke) + Levin group (Q. Chen et al., Chicago), Science Express, doi: /science (Jan 27, 2005) Thomas group (Duke) + Levin group (Q. Chen et al., Chicago), Science Express, doi: /science (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.Leon Cooper[1]Fermi energy

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.

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

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

Tuning interaction via a Feshbach resonance 6 Li

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

Interactions Interaction energy Neglecting the spin indices

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

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

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 40 K ( Jin, JILA; Inguscio, LENS) Mainly 40 K ( Jin, JILA; Inguscio, LENS) and 6 Li ( Hulet, Rice; Salomon, ENS; Thomas, Duke; Ketterle, MIT; Grimm, Innsbruck ) Confined in magnetic and optical traps Confined in magnetic and optical traps Atomic number N= Atomic number N= Fermi temperature E F ~ 1  K Fermi temperature E F ~ 1  K Cooled down to T~ nK Cooled down to T~ nK Two spin mixtures – (pseudo spin) up and down Two spin mixtures – (pseudo spin) up and down Interaction tunable via Feshbach resonances Interaction tunable via Feshbach resonances

Making superfluid condensate with fermions  BEC of diatomic molecules  BCS superconductivity/superfluidity 1. Bind fermions together. 2.BEC 3.Attractive interaction needed Condensation of Cooper pairs of atoms (pairing in momentum space) EFEF spin  spin 

Lecture 2

Physical Picture of BCS-BEC crossover: Tuning the attractive interaction  SC, T C  T*  SC, T C  T* Exists a pseudogap BCSPG/UnitaryBEC Two types of excitations Two types of excitations Change of character: fermionic ! Bosonic Change of character: fermionic ! Bosonic Pairs form at high T Pairs form at high T ( Uc – critical coupling) ( Uc – critical coupling)

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

Terminology Molecules – Feshbach resonance induced molecular bosons – Feshbach molecules – Feshbach bosons --- Should be distinguished from 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 Unitarity – unitary limit -- where a diverges This is the strongly interacting or pseudogap phase (  SC, T C  T* ) BEC limit : BEC limit : Strong attractive interaction – fermions Strong attractive interaction – fermions Weak repulsive interaction – bosons or pairs Weak repulsive interaction – bosons or pairs

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

Evidence for superfluidity 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). T i /T F = Time of flight absorption image

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

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

Highlights of previous work on high Tc  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 Extended ground state crossover to finite T, with a self- consistent treatment of the pseudogap. Q. Chen et al, PRL 81, 4708 (1998)

Highlights of our work on cold atoms The first one that introduced the pseudogap to cold atom physics, calculated Tc, superfluid density, etc 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, (2005)

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

Summary Ultracold Fermi gases near Feshbach resonances are a perfect testing ground for a crossover theory due to tunable interactions. 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. Will help understanding high Tc problem. Signature of superfluidity in the crossover / unitary regime is highly nontrivial. Signature of superfluidity in the crossover / unitary regime is highly nontrivial. We and Duke group have found the strongest evidence for fermionic superfluidity. We and Duke group have found the strongest evidence for fermionic superfluidity. In the process, we developed thermometry. 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: Integrate out boson field: T-matrix t(Q)= T-matrix t(Q)= Fermion self-energy: Fermion self-energy:  2 =  pg 2 +  sc 2

Self-consistent Equations Gap equation: BEC condition Gap equation: BEC condition Number equation: Chemical potential Number equation: Chemical potential Pseudogap equation: Pair density 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 Understanding the profiles at unitarity Theoretical support to TF based thermometry in the strongly interacting regime Uncondensed pairs smooth out the profiles PRL 94, (2005)

Profile decomposition Condensate Noncondensed pairs Fermions

Thermodynamics of Fermi gases Bosonic contribution to thermodynamic potential Bosonic contribution to thermodynamic potential Entropy: fermionic and bosonic. Entropy: fermionic and bosonic.

Entropy of Fermi gases in a trap Power law different from noninteracting Fermi or Bose gases Power law different from noninteracting Fermi or Bose gases Fall in between, power law exponent varies. Fall in between, power law exponent varies. Can be used to determine T for adiabatic field sweep experiments Can be used to determine T for adiabatic field sweep experiments

Thermodynamics of Fermi gases Temperature calibrated to account for imperfection of TF fits Very good quantitative agreement with experiment Science Express, doi: /science (2005) Science Express, doi: /science (2005) Experimental T

Conclusions Interaction between ultracold fermions can be tuned continuously from BCS to BEC. This may eventually shed light on high Tc superconductivity. 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. 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. Our theory works very well in fermionic superfluidity in cold atoms.

A whole new field Interface of AMO and condensed matter physics Excitons in semiconductors Cooper pairs of electrons in superconductors 3 He atom pairs in superfluid 3 He-A,B Neutron pairs, proton pairs in nuclei And neutron stars Mesons in neutron star matter Alkali atoms in ultracold atom gases