Ultracold Quantum Gases Claude Cohen-Tannoudji NCKU, 23 March 2009 Collège de France.

Ultracold Quantum Gases

Claude Cohen-Tannoudji

NCKU, 23 March 2009

Collège de France

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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

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.

Evolution of Atomic Physics

1 – Briefly describe the basic methods used for producing and manipulating ultracold atoms and molecules

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PRODUCING AND MANIPULATINGULTRACOLD ATOMS AND MOLECULES

• Radiative forces

• Cooling

• Trapping

• Feshbach resonances

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Forces exerted by light on atoms

A simple example

Target C bombarded by projectiles p coming all along the same direction

C

p pp

pp

p

p

p

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 incomingphotons, 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 (radiative lifetime of the excited state) of the order of 10-8 s

Mean number of fluorescence cycles per sec : W ~ 1/ ~ 108 sec-1

Stopping an atomic beam

In each cycle, the mean velocity change of the atom is equal to: v = vrec = h/Mc 10-2 m/sMean acceleration a (or deceleration) of the atom

a = velocity change /sec = velocity change v / cycle x number of cycles / sec W = vrec x (1 / R)= 10-2 x 108 m/s2 = 106 m/s2 = 105 g

Huge radiation pressure force!

Atomic beam

Laser beam

Tapered solenoid

J. Prodan W. Phillips H. Metcalf

Zeeman slower

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Laser Doppler coolingT. Hansch, A. Schawlow, D. Wineland, H. Dehmelt

Theory : V. Letokhov, V. Minogin, D. Wineland, W. Itano

2 counterpropagating laser beams

Same intensity Same frequency L (L < A) L < A L < Av

Atom at rest (v=0)

The two radiation pressure forces cancel each other outAtom moving with a velocity v

Because of the Doppler effect, the counterpropagating wave gets closer to resonance and exerts a strongerforce than the copropagating wave which gets fartherNet force opposite to v and proportional to v for v smallFriction force “Optical molasses”

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“Sisyphus” cooling J. DalibardC. 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 K range

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Evaporative cooling

After the collision, the 2 atoms have energiesE3 et E4, with E1+ E2= E3+ E4

Atoms trapped in a potential well with a finite depth U0

2 atoms with energiesE1 et E2 undergo an elastic collision

E4

E2

E1 U0

E3

If E4 > U0, the atom with energy E4 leaves the well

The remaining atom has amuch lower energy E3.After rethermalization of the atoms remaining trapped,the temperature decreases

H. Hess, J.M. Doyle MIT

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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

Spatial gradients of laser intensity

Focused laser beam. Red detuning (L < A)

The light shift Eg 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

Other types of traps using magnetic field gradients combined with the radiation pressure of properly polarized laser beams (“Magneto Optical Traps”)

“Optical Tweezers”

“Optical lattice”

Spatially periodic array of potential wells associated with the light shifts of a detuned laser standing wave

A. Ashkin, S. Chu

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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 latticesThey can be easily manipulated, much more than the periodicpotential inside a crystal

Furthermore, possibility to control atom-atom interactions, both in magnitude and sign, by using “Feshbach resonances”

- 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 frequency of one of the 2 waves and to obtain a moving standing wave

- Possibility to change the spatial period of the potential by changing the angle between the 2 running laser waves

- Possibility to change the dimensionality (1D, 2D, 3D) and the symmetry (triangular lattice, cubic lattice)

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Closed channel

Open channel

E

r

V

0

Feshbach Resonances

The 2 atoms collide with a very small positive energy E in a channel which is called “open”

The energy of the dissociation threshold of the open channel is taken as the zero of energy

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

Ebound

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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 h can be absorbed by an atom which is brought to an excited discrete state with an energy h0 above the initial atomic state and then reemitted. There is a resonance in the scattering amplitude when is close to 0

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Closed channel

Open channel

E

r

V

0

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

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0B0bga

Scattering length versus magnetic field

a > 0Repulsive effective long

range interactions

a < 0Attractive effective long

range interactions

a = 0No interactions

Perfect gas

Near B=B0, IaI is very largeStrong 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

abg

Backgroundscattering

length

a

B

First observation for cold Na atoms: MIT Nature, 392, 151 (1998)

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B

Eb

0B

The bound state exists only in the region a > 0. It has a spatial extension a and an energy Eb= - ħ2 / ma2

a > 0Bound state with an energy

Eb= - ħ2 / ma2 - (B – B0)2

a < 0No bound state

Bound state of the two-channel Hamiltonian

a =

Weakly bound dimer with universal properties Quantum “halo” state or “Feshbach molecule”

In the region a » range r0 of atom–atom interactions

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B

Eb

0B

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

Formation of a Fehbach molecule

a > 0Bound state with an energy

Eb= - ħ2 / ma2 - B2

a < 0No bound state

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-photonphotoassociation

E

r

A+A

A+A*

One-photon PATwo-photon PA

Recent results obtained in Paris on the PA of two metastable helium atomswith a high internal energy

Giant dimmers produced by one-photon PA Distance between the 2 atoms larger than 50 nmNeed to include retardation effects in the Van der Waals interactions

for explaining the vibrational spectrumMolecules 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 LAWSWITH ULTRACOLD ATOMS

Ultraprecise Atomic Clocks

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Interrogation Correction

Oscillator 0

Atomic transition

1 / T T : Observation time

The narrower the atomic line,i.e. the smaller the better the locking of the frequency of the oscillator to A.

It is therefore interesting to use slow atoms in order to increase T, and thus to decrease

The correction loop locks thefrequency of the oscillator tothe frequency A of the hyperfinetransition of 133Cs used for definingthe second

Principle of an atomic clock

Measuring time with atomic clocks

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Improving atomic clocks with ultracold atoms

Usual clocks using thermal Cs atoms

Cs atomic beamv 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

H

Fountains of ultracold atomsThrowing 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!

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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

Examples of atomic fountains

- Sodium fountains : Stanford S. Chu- Cesium fountains : BNM/SYRTE C. Salomon, A. Clairon

ChristopheSalomon

AndréClairon

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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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•Thermal beam : v = 100 m/s, T = 5 ms = 100 Hz

•Fountain : v = 4 m/s, T = 0.5 s = 1 Hz

•PHARAO : v = 0.05 m/s, T = 5 s = 0.1 Hz

Sensitivity gainsSensitivity gains

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ACES on the space stationACES on the space station cnesesa

• 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 shiftof the frequency of a clock

An observer at an altitude z receives the signal of a clock located at the altitude z+z and measures a frequency A(z+z) different from the frequency, A(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 testsAnother possible application : determination of the “geoid”, surface where the gravitational potential has a given value

Year

Rel

ativ

e ac

cura

cy

Optical clocks

Cs Clocks

Atomic fountains

Redefinition of the second

Combs

Recent results obtained by the NIST-Boulder group

Science, 319, 1808 (2008)

Single ion optical clocks with Al+ and Hg+

Tests of a possible variation of fundamental constants

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FROM ULTRACOLD ATOMSTO 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

Ultracold atoms as “quantum simulators”

Fermionic atoms in an optical lattice - “Metal” Mott-insulator transition - Towards antiferromagnetic structures

All atoms are in the same All atoms are in the same quantum state and evolve quantum state and evolve in phase like soldiers in phase like soldiers marching in loskstepmarching in loskstep

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 themThey form a macroscopic matter wave

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)

JILA 87Rb1995

MIT23Na1995

Experimental observation

Many others atoms have been condensed 7Li, 1H, 4He*, 41K, 133Cs, 174Yb, 52Cr…

Atom lasers

Lattice of quantized vortices in a condensate

Lattice of quantized vortices in a superconductor

Superfluidity

Coherence

Examples of quantum propertiesof macroscopic matter waves of bosonic atoms

Interferencesbetween 2 condensates

MIT Coherent beam of atomic de Broglie waves extracted

from a condensate

Munich

ENSMIT

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BEC in a periodic optical potential Superfluid – Mott insulator transition

a – Small depth of the wells. Delocalized matter waves. Superfluid phase

a b

b - Large depth of the wells. Localized waves. Insulator phase

a b a

I. Bloch groupin MunichNature,415, 39 (2002)

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 quantizedvortices in all these 3 zonesdemonstrating the superfluidcharacter 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

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

0 0

J. Dalibard group, ENS, Paris, Nature, 441,1053 (2006)

Conclusion : the BKT crossover is due to the unbinding ofvortex-antivortex pairs with the appearance of free vortices

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- Mixture in equal proportions of fermionic atoms in 2 different states in an optical lattice

Spin up: Spin down:

Fermionic Mott insulator

- 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)

Non interacting fermions (single band model)

Compressible “metal”

Compression

Band insulator

Repulsive interactions

Compression

Compressible “metal” Mott insulator

Clo

ud s

ize

(com

pre

ssib

ility

)

2N 2/3

MIBI

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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 « )

Antiferromagnetic order in a square lattice

or ?

Triangular lattice Frustration

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ConclusionUsing 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

• 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

Requirements for a “quantum simulator”

• 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

Possibilities offered by ultracold atomic gases

Hope to answer in this way questions unreachable for classicalcomputers because of memory, speed and size limitations

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