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 Quantum degenerate particles: fermions vs particles: fermions vs

bosonsbosons

Bose-Einstein condensation Fermi sea of atoms

EF = kBTF

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 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 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

BEC in Bosonic Alkali BEC in Bosonic Alkali AtomsAtoms

BEC – Fifth state of matterBEC – Fifth state of matter BEC was predicted in 1924BEC was predicted in 1924 Achieved in dilute gases of alkali atoms in Achieved in dilute gases of alkali atoms in

19951995 Nobel Prize in physics 2001: Nobel Prize in physics 2001:

Eric A Cornell, Carl E Wieman, Wolfgang Ketterle 87Rb

400 nK

200 nK

50 nK

Momentum distribution

Momentum distribution of a BEC

400 nK

200 nK

50 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 Physics of BEC – Bose StatisticsStatistics

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 Superfluidity in Fermi SystemsSystems

Theory of superconductivity, 1957Theory of superconductivity, 1957 J. Bardeen, L.N. Cooper, J.R. Schrieffer J. Bardeen, L.N. Cooper, J.R. Schrieffer Nobel Prize -1972Nobel Prize -1972

Discovery of superfluid Discovery of superfluid 33He, He, 19721972

D.M. Lee, D.D. Osheroff D.M. Lee, D.D. Osheroff R.C. RichardsonR.C. Richardson

Nobel Prize - 1996Nobel Prize - 1996

Discovery of Discovery of superconductivity, 1911superconductivity, 1911

Heike Kamerlingh Onnes Heike Kamerlingh Onnes Nobel Prize -1913Nobel Prize -1913

High Tc superconductors, High Tc superconductors, 19861986

J.G. Bednorz, K.A. MüllerJ.G. Bednorz, K.A. Müller Nobel Prize - 1987Nobel Prize - 1987 Nobel Prize in physics Nobel Prize in physics

20032003 A.A. Abrikosov (vortex A.A. Abrikosov (vortex lattice)lattice) V.L. Ginzburg (LG theory)V.L. Ginzburg (LG theory) A.J. Leggett (superfluid A.J. Leggett (superfluid 33He)He)

Where will the next Nobel Prize be?

Superfluidity in Atomic Superfluidity in Atomic Fermi GasesFermi Gases

Quantum degenerate atomic Fermi gas – 1999Quantum 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 – 2003Creation of bound di-atomic molecules – 2003 4040KK: Jin group (JILA), Nature : Jin group (JILA), Nature 424424, 47 (2003). , 47 (2003). 66LiLi: Hulet group (Rice),: Hulet group (Rice), PRL 91, 080406 (2003); PRL 91, 080406 (2003); 66LiLi: Grimm group (Innsbruck), PRL 91, 240402 (2003): Grimm group (Innsbruck), PRL 91, 240402 (2003)

Molecular BEC from atomic Fermi gases – Nov Molecular BEC from atomic Fermi gases – Nov 20032003 4040KK: Jin group, Nature : Jin group, Nature 426426, 537 (2003). , 537 (2003). 66LiLi: Grimm group, Science 302, 2101 (2003): Grimm group, Science 302, 2101 (2003) 66LiLi: Ketterle group (MIT), PRL : Ketterle group (MIT), PRL 9191, 250401 (2003)., 250401 (2003).

Fermionic superfluidity Fermionic superfluidity (Cooper pairs) – 2004(Cooper pairs) – 2004 Jin group, PRL 92, 040403 (2004)Jin group, PRL 92, 040403 (2004) Grimm group, Science 305, 1128 (2004)Grimm group, Science 305, 1128 (2004) Ketterle group, PRL Ketterle group, PRL 9292, 120403 (2004)., 120403 (2004).

Superfluidity in Atomic Superfluidity in Atomic Fermi GasesFermi Gases

Molecular BEC from atomic Fermi gases – Molecular BEC from atomic Fermi gases – Nov 2003Nov 2003 4040KK: Jin group, Nature : Jin group, Nature 426426, 537 (2003). , 537 (2003). 66LiLi: Grimm group, Science 302, 2101 (2003): Grimm group, Science 302, 2101 (2003) 66LiLi: Ketterle group (MIT), PRL : Ketterle group (MIT), PRL 9191, 250401 (2003)., 250401 (2003).

Fermionic superfluidity Fermionic superfluidity (Cooper pairs) – 2004(Cooper pairs) – 2004 Jin group, PRL 92, 040403 (2004)Jin group, PRL 92, 040403 (2004) Grimm group, Science 305, 1128 (2004)Grimm group, Science 305, 1128 (2004) Ketterle group, PRL Ketterle group, PRL 9292, 120403 (2004)., 120403 (2004).

Heat capacity measurement + Heat capacity measurement + thermometry in strongly interacting thermometry in strongly interacting regime – 2004regime – 2004

Thomas group (Duke) + Levin group (Thomas group (Duke) + Levin group (Q. ChenQ. Chen et et al.al., Chicago), Science Express, , Chicago), Science Express, doi:10.1126/science.1109220 (Jan 27, 2005)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, 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>

Tuning Tuning interactioninteraction in atoms via in atoms via a Feshbach resonancea Feshbach resonance

R R

a<0, weak attractiona>0, strong attractionbound state

V(R)

R R

We can control attraction via B field !

Tuning interaction via a Tuning interaction via a Feshbach resonanceFeshbach resonance

06Li

Introduction to BCS Introduction to BCS theorytheory

22ndnd quantization – quantum field quantization – quantum field theory – many-body theorytheory – many-body theory

Fermi gasesFermi 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 theoryOverview of BCS theoryFermi Gas

No excitation gap

BCS superconductor

BCS theory works very well BCS theory works very well for weak coupling for weak coupling superconductorssuperconductors

Facts About Trapped Fermi Facts About Trapped Fermi Gases Gases

Mainly Mainly 4040K (K (Jin, JILA;Jin, JILA; Inguscio, LENS) Inguscio, LENS)

and and 66Li (Li (Hulet, Rice; Salomon, ENS; Thomas, Hulet, Rice; Salomon, ENS; Thomas, Duke; Ketterle, MIT; Grimm, InnsbruckDuke; Ketterle, MIT; Grimm, Innsbruck))

Confined in magnetic and optical trapsConfined in magnetic and optical traps Atomic number N=10Atomic number N=1055-10-1066

Fermi temperature Fermi temperature EEFF ~ 1 ~ 1 KK Cooled down to Cooled down to TT~10-100 nK~10-100 nK Two spin mixtures – (pseudo spin) up and Two spin mixtures – (pseudo spin) up and

downdown Interaction tunable via Feshbach Interaction tunable via Feshbach

resonances resonances

Making superfluid condensate Making superfluid condensate with fermions with fermions

BEC of diatomic molecules

BCS superconductivity/superfluidity

1. Bind fermions together.2. BEC3. Attractive interaction needed

Condensation of Cooper pairs of atoms

(pairing in momentum space)

EF

spin spin

Lecture 2Lecture 2

Physical Picture of BCS-BEC Physical Picture of BCS-BEC crossover:crossover:

Tuning the attractive Tuning the attractive interactioninteraction

SC SC , T, TCC T* T*Exists a pseudogapExists a pseudogap

BCS PG/Unitary BEC

Two types of Two types of excitationsexcitations

Change of Change of character: character: fermionic fermionic !! BosonicBosonic Pairs form at Pairs form at high high TT ((UcUc – critical – critical coupling)coupling)

High Tc superconductors: High Tc superconductors: Tuning parameter: hole doping Tuning parameter: hole doping

concentration concentration

Increasing interaction

Cannot reach bosonic regime due to d-wave pairing

Crossover and pseudogap Crossover and pseudogap physics in high Tc physics in high Tc superconductorssuperconductors

BCS-BEC crossover provides a natural BCS-BEC crossover provides a natural explanation for the PG phenomena. explanation for the PG phenomena.

Q. Chen, I. Kosztin, B. Janko, and K. Levin, PRL Q. Chen, I. Kosztin, B. Janko, and K. Levin, PRL 81, 4708 (1998)81, 4708 (1998)

BSCCO, H. Ding et al, Nature 1996

Crossover under control in Crossover under control in cold Fermi atoms (cold Fermi atoms (11stst time time

possiblepossible))

Molecules of fermionic atoms

BEC of bound molecules

Cooper pairs

BCS superconductivityCooper pairs: correlated momentum-space pairing

kF

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

TerminologyTerminology Molecules – Feshbach resonance induced Molecules – Feshbach resonance induced

molecular bosons – Feshbach molecules – molecular bosons – Feshbach molecules – Feshbach bosons --- Feshbach bosons --- Should be distinguished Should be distinguished fromfromCooper pairsCooper pairs -- many-body effect induced giant -- many-body effect induced giant pairspairs

Unitarity – unitary limit -- where Unitarity – unitary limit -- where aa divergesdivergesThis is the strongly interacting or pseudogap This is the strongly interacting or pseudogap phase phase ((SC SC , T, TCC T* T* ) )

BEC limit : BEC limit : Strong attractive interaction – fermionsStrong attractive interaction – fermions Weak repulsive interaction – bosons or pairsWeak repulsive interaction – bosons or pairs

Big questions –Big questions –

Cold atoms may help understanding Cold atoms may help understanding high Tchigh Tc

How to determine whether the system How to determine whether the system is in the superfluid phase? is in the superfluid phase? Charge neutralCharge neutral Existence of pseudogapExistence of pseudogap

How to measure the temperature?How to measure the temperature? Most interesting is the Most interesting is the

pseudogap/unitary regime – diverging pseudogap/unitary regime – diverging scattering length – strongly interactingscattering length – strongly interacting

Evidence for Evidence for superfluiditysuperfluidity

Molecular Molecular CondensateCondensateBimodal density Bimodal density distributiondistribution

Adiabatic/slow Adiabatic/slow sweep from BCS sweep from BCS side to BEC side. side to BEC side. Molecules form Molecules form and Bose and Bose condense.condense. M. Greiner, C.A. Regal, and D.S. Jin, Nature 426, 537 (2003).

Ti/TF = 0.19 0.06Time of flightabsorption image

-0.5 0.0 0.5

0

1x105

2x105

3x105

N m

olec

ules

Cooper pair condensateCooper pair condensate

C. Regal, M. Greiner, and D. S. Jin, PRL 92, 040403 (2004)

Dissociation of moleculesat low density

B = 0.12 G B = 0.25 G B=0.55 G

T/TF=0.08

B (gauss)

Observation of pseudogapObservation of pseudogap-Pairing gap measurements using RF -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 Highlights of previous work on high Tcon high Tc

Phase diagram for high Tc Phase diagram for high Tc superconductors, in superconductors, in (semi-) quantitative (semi-) quantitative agreement with agreement with experiment.experiment.

Quasi-universal behavior Quasi-universal behavior of superfluid density.of superfluid density.

The only one in high Tc The only one in high Tc that is capable of that is capable of quantitative calculationsquantitative calculations

We are now in a We are now in a position to work on position to work on cold atomscold atoms

Extended ground state crossover to finite T, with a self- consistent treatment of the pseudogap.

Q. Chen Q. Chen et alet al, PRL 81, 4708 (1998), PRL 81, 4708 (1998)

Highlights of our work on Highlights of our work on cold atomscold atoms

The first one that introduced the pseudogap to cold atom The first one that introduced the pseudogap to cold atom physics, calculated Tc, superfluid density, etcphysics, calculated Tc, superfluid density, etc

Signatures of superfluidity and understanding density profiles

PRL 94, 060401 (2005)

Highlights of our work on Highlights of our work on cold atomscold atoms

First evidence (with experiment) for a superfluid First evidence (with experiment) for a superfluid phase transition phase transition

Thermodynamic properties of strongly interacting trapped gases

Science Express, doi:10.1126/science.1109220 (2005)Science Express, doi:10.1126/science.1109220 (2005)

SummarySummary

Ultracold Fermi gases near Feshbach Ultracold Fermi gases near Feshbach resonances are a perfect testing ground for a resonances are a perfect testing ground for a crossover theory due to tunable interactions.crossover theory due to tunable interactions.

Will help understanding high Tc problem.Will help understanding high Tc problem. Signature of superfluidity in the crossover / Signature of superfluidity in the crossover /

unitary regime is highly nontrivial.unitary regime is highly nontrivial. We and Duke group have found the strongest We and Duke group have found the strongest

evidence for fermionic superfluidity.evidence for fermionic superfluidity. In the process, we developed thermometry.In the process, we developed thermometry.

Theoretical Formalism and Results

Grand canonical Grand canonical Hamiltonian for resonance Hamiltonian for resonance

superfluiditysuperfluidity

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 formalismT-matrix formalism Integrate out boson field:Integrate out boson field:

T-matrix t(Q)= T-matrix t(Q)=

Fermion self-energyFermion self-energy::

2 = pg2 + sc

2

Self-consistent EquationsSelf-consistent Equations

Gap equationGap equation: BEC condition: BEC condition

Number equationNumber equation: Chemical : Chemical

potentialpotential

Pseudogap equationPseudogap equation: Pair density: Pair density

Critical temperatureCritical temperatureHomogeneous 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 Understanding the profiles at unitarityunitarity

• Theoretical support to TF based thermometry in the strongly interacting regime

Uncondensed pairs smooth out the profiles

PRL 94, 060401 (2005)

Profile decompositionProfile decomposition

Condensate Noncondensed pairs Fermions

Thermodynamics of Thermodynamics of Fermi gasesFermi gases

Bosonic Bosonic contribution to contribution to thermodynamic thermodynamic potentialpotential

Entropy: Entropy: fermionic and fermionic and bosonic.bosonic.

Entropy of Fermi gases in Entropy of Fermi gases in a trapa trap

Power law Power law different from different from noninteracting noninteracting Fermi or Bose Fermi or Bose gasesgases

Fall in between, Fall in between, power law power law exponent varies.exponent varies.

Can be used to Can be used to determine determine TT for for adiabatic field adiabatic field sweep sweep experimentsexperiments

Thermodynamics of Fermi Thermodynamics of Fermi gasesgases

Temperature calibrated to

account for imperfection of TF fits

• Very good quantitative agreement with experiment

Science Express, Science Express, doi:10.1126/science.1109220 (2005)doi:10.1126/science.1109220 (2005)

Experimental T

ConclusionsConclusions

Interaction between ultracold fermions Interaction between ultracold fermions can be tuned continuously from BCS to can be tuned continuously from BCS to BEC. This may eventually shed light on BEC. This may eventually shed light on high Tc superconductivity.high Tc superconductivity.

Except in the BCS regime, opening of Except in the BCS regime, opening of an excitation gap can no longer be an excitation gap can no longer be taken as a signature of superfluidity. taken as a signature of superfluidity. Pseudogap makes these gases more Pseudogap makes these gases more complicated and interesting. complicated and interesting.

Our theory works very well in Our theory works very well in fermionic superfluidity in cold atoms.fermionic superfluidity in cold atoms.

A whole new A whole new fieldfield

Interface of AMO and condensed matter Interface of AMO and condensed matter physicsphysics

Excitons insemiconductors

Cooper pairs of electrons in

superconductors

3He atom pairs in superfluid

3He-A,B

Neutron pairs, proton pairs in nuclei

And neutron stars

Mesons in neutron star

matter

Alkali atomsin ultracold atom gases