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Ultracold Quantum Gases
Claude Cohen-Tannoudji
NCKU, 23 March 2009
Collège de France
Evolution of Atomic Physics
Characterized by spectacular advances in our ability to
manipulate the various degrees of freedom of an atom
- Spin polarization (optical pumping)
- Velocity (laser cooling, evaporative cooling)
- Position (trapping)
- Atom-Atom interactions (Feshbach resonances)
Purpose of this lecture
1 – Briefly describe the basic methods used for producing and
manipulating ultracold atoms and molecules
2 - Review a few examples showing how ultracold atoms are
allowing one to
 perform new more refined tests of basic physical laws
 achieve new situations where all parameters can be carefully
controlled, providing in this way simple models for
understanding more complex problems in other fields.
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PRODUCING AND MANIPULATING
ULTRACOLD ATOMS AND MOLECULES
• Radiative forces
• Cooling
• Trapping
• Feshbach resonances
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Forces exerted by light on atoms
A simple example
p
p
Target C bombarded by
projectiles p coming all
along the same direction
p
p
p
p
p
p
C
As a result of the transfer of momentum from the
projectiles to the target C, the target C is pushed
Atom in a light beam
Analogous situation, the incoming
photons, scattered by the atom C
playing the role of the projectiles p
Explanation of the tail of the comets
In a resonant laser beam, the radiation
pressure force can be very large
Sun
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Atom in a resonant laser beam
Fluorescence cycles (absorption + spontaneous emission) lasting
a time t (radiative lifetime of the excited state) of the order of 10-8 s
Mean number of fluorescence cycles per sec : W ~ 1/ t ~ 108 sec-1
In each cycle, the mean velocity change of the atom is equal to:
dv = vrec = hn/Mc  10-2 m/s
Mean acceleration a (or deceleration) of the atom
a = velocity change /sec
= velocity change dv / cycle x number of cycles / sec W
= vrec x (1 / tR)= 10-2 x 108 m/s2 = 106 m/s2 = 105 g
Huge radiation pressure force!
Stopping an atomic beam
Laser
beam
Atomic
beam
Tapered solenoid
Zeeman slower
J. Prodan
W. Phillips
H. Metcalf
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Laser Doppler cooling
T. Hansch, A. Schawlow, D. Wineland, H. Dehmelt
Theory : V. Letokhov, V. Minogin, D. Wineland, W. Itano
2 counterpropagating laser beams
Same intensity
Same frequency nL (nL < nA)
nL < nA
nL < nA
v
Atom at rest (v=0)
The two radiation pressure forces cancel each other out
Atom moving with a velocity v
Because of the Doppler effect, the counterpropagating
wave gets closer to resonance and exerts a stronger
force than the copropagating wave which gets farther
Net force opposite to v and
proportional to v for v small
Friction force “Optical molasses”
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“Sisyphus” cooling
J. Dalibard
C. Cohen-Tannoudji
Several ground state sublevels
Spin up
Spin down
In a laser standing wave, spatial modulation of the laser intensity and
of the laser polarization
• Spatially modulated light shifts of g and g due to the laser light
• Correlated spatial modulations of optical pumping rates g ↔ g
The moving atom is always running up potential hills (like Sisyphus)!
Very efficient cooling scheme leading to temperatures in the mK range
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Evaporative cooling
H. Hess, J.M. Doyle
MIT
E4
E2
E1
U0
E3
Atoms trapped in a
potential well with
a finite depth U0
2 atoms with energies
E1 et E2 undergo an
elastic collision
After the collision, the
2 atoms have energies
E3 et E4, with
E1+ E2= E3+ E4
If E4 > U0, the atom with
energy E4 leaves the well
The remaining atom has a
much lower energy E3.
After rethermalization of the
atoms remaining trapped,
the temperature decreases
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Temperature scale (in Kelvin units)
cosmic microwave background radiation
(remnant of the big bang)
The coldest matter in the universe
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Traps for neutral atoms
“Optical Tweezers”
A. Ashkin, S. Chu
Spatial gradients of laser intensity
Focused laser beam. Red detuning (wL < wA)
The light shift dEg of the ground state g is negative and
reaches its largest value at the focus. Attractive potential well
in which neutral atoms can be trapped if they are slow enough
“Optical lattice”
Spatially periodic array of potential
wells associated with the light shifts
of a detuned laser standing wave
Other types of traps using magnetic field gradients combined
with the radiation pressure of properly polarized laser beams
(“Magneto Optical Traps”)
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Optical lattices
The dynamics of an atom in a periodic optical potential, called
“optical lattice”, shares many features with the dynamics of an
electron in a crystal. But it also offers new possibilities!
New possibilities offered by optical lattices
They can be easily manipulated, much more than the periodic
potential inside a crystal
- Possibility to switch off suddenly the optical potential
- Possibility to vary the depth of the periodic potential well
by changing the laser intensity
- Possibility to change the spatial period of the potential by
changing the angle between the 2 running laser waves
- Possibility to change the frequency of one of the 2 waves
and to obtain a moving standing wave
- Possibility to change the dimensionality (1D, 2D, 3D) and
the symmetry (triangular lattice, cubic lattice)
Furthermore, possibility to control atom-atom interactions, both
in magnitude and sign, by using “Feshbach resonances”
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Feshbach Resonances
The 2 atoms collide with a very
small positive energy E in a
channel which is called “open”
V
The energy of the dissociation
threshold of the open channel is
taken as the zero of energy
Closed
channel
Ebound
E
0
r
Open
channel
There is another channel above
the open channel where
scattering states with energy E
cannot exist because E is below
the dissociation threshold of this
channel which is called “closed”
There is a bound state in the
closed channel whose energy
Ebound is close to the collision
energy E in the open channel
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Physical mechanism of the Feshbach resonance
The incoming state with energy E of the 2 colliding atoms in
the open channel is coupled by the interaction to the bound
state bound in the closed channel.
The pair of colliding atoms can make a virtual transition to the
bound state and come back to the colliding state. The duration
of this virtual transition scales as ħ / I Ebound-E I, i.e. as the
inverse of the detuning between the collision energy E and the
energy Ebound of the bound state.
When E is close to Ebound, the virtual transition can last a very
long time and this enhances the scattering amplitude
Analogy with resonant light scattering when an impinging
photon of energy hn can be absorbed by an atom which is
brought to an excited discrete state with an energy hn0 above
the initial atomic state and then reemitted. There is a
resonance in the scattering amplitude when n is close to n0
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Sweeping the Feshbach resonance
The total magnetic moment of the atoms are not the same in the 2
channels (different spin configurations). The energy difference
between the them can be varied by sweeping a magnetic field
V
Closed
channel
E
0
r
Open
channel
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Scattering length versus magnetic field
a
a>0
Repulsive effective long
range interactions
0
B0
abg
abg
Background
scattering
length
a<0
Attractive effective long
range interactions
a=0
No interactions
Perfect gas
B
Near B=B0, IaI is very large
Strong interactions
Strong correlations
B0 : value of B for which the energy of the bound state, in the closed channel
(shifted by its interaction with the continuum of collision states in the open
channel) coincides with the energy E~0 of the colliding pair of atoms
First observation for cold Na atoms:
MIT Nature, 392, 151 (1998)
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Bound state of the two-channel Hamiltonian
In the region a » range r0 of atom–atom interactions
Eb
a =
a<0
No bound state
a>0
Bound state with an energy
Eb= - ħ2 / ma2  - (B – B0)2
B0
B
The bound state exists only in the region a > 0. It has
a spatial extension a and an energy Eb= - ħ2 / ma2
Weakly bound dimer with universal properties
Quantum “halo” state or “Feshbach molecule”
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Formation of a Fehbach molecule
Eb
a>0
a<0
Bound state with an energy
Eb= - ħ2 / ma2  - B2
No bound state
B0
B
If B0 is swept through the Feshbach resonance from the region
a < 0 to the region a > 0, a pair of colliding ultracold atoms can
be transformed into a Feshbach molecule
Another interesting system: Efimov trimers (R. Grimm)
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Another method for producing ultracold Molecules
Gluing 2 ultracold atoms
with one or two-photon
photoassociation
Recent results obtained in
Paris on the PA of two
metastable helium atoms
with a high internal energy
E
A+A*
One-photon PA
Two-photon PA
A+A
r
Giant dimmers produced by one-photon PA
Distance between the 2 atoms larger than 50 nm
Need to include retardation effects in the Van der Waals interactions
for explaining the vibrational spectrum
Molecules of metastable He produced by two-photon PA
Measurement of the binding energy of the least bound state and
determination of the scattering length of 2 metastable He atoms
with an accuracy more than 100 larger than all prevous
measurements
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TESTING FUNDAMENTAL LAWS
WITH ULTRACOLD ATOMS
Ultraprecise Atomic Clocks
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Measuring time with atomic clocks
Principle of an atomic clock
The correction loop locks the
frequency of the oscillator to
the frequency wA of the hyperfine
transition of 133Cs used for defining
Atomic transition
the second
The narrower the atomic line,
i.e. the smaller Dw , the better Interrogation
Correction
the locking of the frequency of
the oscillator to wA.
Dw1/T
Oscillator w0
T : Observation time
It is therefore interesting to use slow atoms in order to
increase T, and thus to decrease D w
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Improving atomic clocks with ultracold atoms
Usual clocks using thermal Cs atoms
Cs atomic beam
v  100 m/s
ℓ
ℓ
L  0.5 m
Appearance in the resonance of Ramsey fringes having a
width determined by the time T = L / v  0.005 s
Fountains of ultracold atoms
Throwing a cloud of ultracold atoms upwards
with a laser pulse to have them crossing the
same cavity twice, once in the way up, once in
the way down, and obtaining in this way 2
interactions separated by a time interval T
H = 30 cm  T = 0.5 s
Improvement by a factor 100!
H
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Examples of atomic fountains
- Sodium fountains :
- Cesium fountains :
Christophe
Salomon
Stanford S. Chu
BNM/SYRTE
C. Salomon, A. Clairon
André
Clairon
Stability : 1.6 x 10-16 for an integration time 5 x 104 s
Accuracy : 3 x 10-16
A stability of 10-16 corresponds to an error smaller than
1 second in 300 millions years
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From terrestrial clocks to space clocks
Working in microgravity in order to avoid the fall of atoms.
One can then launch them through 2 cavities with a very
small velocity without having them falling
Parabolic flights (PHARAO project)
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Sensitivity gains
• Thermal beam :
v = 100 m/s, T = 5 ms
Dn = 100 Hz
• Fountain :
v = 4 m/s, T = 0.5 s
Dn = 1 Hz
• PHARAO :
v = 0.05 m/s, T = 5 s
Dn = 0.1 Hz
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ACES on the space station
cnes
esa
• Time reference
• Validation of spatial clocks
• Fundamental tests
C. Salomon et al , C. R. Acad. Sci. Paris, t.2, Série IV, p. 1313-1330 (2001)
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Gravitational shift
of the frequency of a clock
An observer at an altitude z receives the signal of a clock
located at the altitude z+dz and measures a frequency
wA(z+dz) different from the frequency, wA(z), of his own clock
2 clocks at altitudes differing by 1 meter have apparent
frequencies which differ in relative value by 10-16.
A space clock at an altitude of 400 kms differs from a
terrestrial clock by 4 x 10-11 . Possibility to check this effect
with a precision 25 times better than all previous tests
Another possible application : determination of the “geoid”,
surface where the gravitational potential has a given value
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Relative accuracy
Optical clocks
Combs
Redefinition of the second
Cs Clocks
Atomic fountains
Year
Recent results obtained by the NIST-Boulder group
Single ion optical clocks with Al+ and Hg+
Tests of a possible variation of fundamental constants
Science,
319, 1808
(2008)
FROM ULTRACOLD ATOMS
TO MORE COMPLEX SYSTEMS
Bose Einstein condensates
Phase transitions involving bosonic atoms or molecules
- Superfluid Mott-insulator transition
- BEC – BCS crossover. From a molecular BEC to a
BCS superfluid of Cooper type pairs of fermionic atoms
- Berezinski-Kosterlitz-Thouless transition for a
two-dimensional Bose gas
Fermionic atoms in an optical lattice
- “Metal” Mott-insulator transition
- Towards antiferromagnetic structures
Ultracold atoms as “quantum simulators”
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Bose Einstein condensates
When T decreases, the de
Broglie wavelength
increases and the size of
the atomic wave packets
increases
When they overlap all
atoms condense in the
ground state of the trap
which contains them
They form a macroscopic
matter wave
All atoms are in the same
quantum state and evolve
in phase like soldiers
marching in loskstep
These gaseous condensates, discovered in 1995, are macroscopic
quantum systems having properties (superfluidity, coherence) which
make them similar to other systems only found up to now in dense
systems (liquid helium , superconductors)
Experimental observation
JILA
87Rb
1995
Many others atoms have been condensed
7Li, 1H, 4He*, 41K, 133Cs, 174Yb, 52Cr…
MIT
23Na
1995
Examples of quantum properties
of macroscopic matter waves of bosonic atoms
Coherence
Atom lasers
Interferences
between 2 condensates
MIT
Munich
Coherent beam of
atomic de Broglie
waves extracted
from a condensate
Superfluidity
ENS
Lattice of quantized
vortices in a condensate
MIT
Lattice of quantized
vortices in a superconductor
BEC in a periodic optical potential
Superfluid – Mott insulator transition
a
a – Small depth of the wells. Delocalized matter waves.
Superfluid phase
b
b - Large depth of the wells. Localized waves. Insulator phase
I. Bloch group
in Munich
Nature,
415, 39 (2002)
a
b
a
Realization of the Bose Hubbard Hamiltonian
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BEC-BCS crossover observed with ultracold fermions
By varying the magnetic field around a Feshbach resonance,
one can explore 3 regions
- Region a>0 (strong interactions). There is a bound state in the
interaction potential where 2 fermions with different spin states
can form molecules which can condense in a molecular BEC
- Region a<0 (weak interactions). No molecular state, but long
range attractive interactions giving rise to weakly bound
Cooper pairs which can condense in a BCS superfluid phase
- Region a= (Very strong interactions)
Strongly correlated systems with universal properties.
Recent observation at MIT
(W. Ketterle et al) of quantized
vortices in all these 3 zones
demonstrating the superfluid
character of the 3 phases
Science, 435, 1047 (2005)
a>0
a=
a<0
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BKT crossover in a trapped 2D atomic gas
J. Dalibard group, ENS, Paris, Nature, 441,1053 (2006)
How to prepare the 2D gas
How to detect phase coherence
Interference fringes changing at high T (lower contrast, waviness)
Quasi-long-range order (vortex-antivortex pairs) lost at high T
Detection of the appearance of free vortices
Onset of sharp dislocations in the
interference pattern coinciding with
the loss of long-range order

0
Conclusion : the BKT crossover is due to the unbinding of
vortex-antivortex pairs with the appearance of free vortices
0
0
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Fermionic Mott insulator
- Mixture in equal proportions of fermionic atoms in 2 different
states in an optical lattice
Spin up:
Spin down:
- Adding an external harmonic confinement pushing the atoms
towards the center of the lattice
- How are the atoms moving in the lattice when their interactions,
the lattice depth, the external confinement are varied
-Competition between
■ Pauli exclusion priciple preventing 2 atoms in the same spin
state to occupy the same lattice site
■ Interactions between atoms in different spin states. If they
are repulsive, the 2 atoms don’t like to be in the same site
■ External confinement
Realization of the Fermi Hubbard Hamiltonian
Two recent experiments : Zurich (ETH) Nature 455, 204 (2008)
Mainz Science 322, 1520 (2008)
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Non interacting fermions
(single band model)
Compression
Compressible “metal”
Band insulator
Repulsive interactions
Compression
Cloud size
(compressibility)
Compressible “metal”
Mott insulator
MI
BI
w2N 2/3
Towards antiferromagnetic structures
Realizing an interaction
- Atoms with a large magnetic dipole (Cr)
- Heteropolar molecules in the ground state
- Super-exchange (Pauli principle + on site interactions)
Need of a very low temperature (kBT « a)
or
Antiferromagnetic order
in a square lattice
Triangular lattice
?
Frustration
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Conclusion
Using ultracold atoms as quantum simulators
Quantum simulator: experimental system whose behavior
reproduces as close as possible a certain class of model
Hamiltonians.
Feynman’s idea
Requirements for a “quantum simulator”
• Tailoring the potential in which particles are moving
• Controlling the interactions between particles
• Controlling the temperature, the density
• Ability to measure various properties of the system
Possibilities offered by ultracold atomic gases
• Very flexible optical potentials, with all dimensionalities,
with all possible shapes (periodic, single well,…)
• Tuning the interactions with Feshbach resonances
• Various cooling schemes and measurement methods
Hope to answer in this way questions unreachable for classical
computers because of memory, speed and size limitations
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