Download Ultracold Atomic Gases

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 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 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)


Where
will the next Nobel Prize be?
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
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 (D  DSC , TC  T* )

BEC limit :
Strong attractive interaction – fermions
 Weak repulsive interaction – bosons or pairs

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 previous work 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
We are now in a position to
work on cold atoms
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
 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