Ultracold Quantum Gases Claude CohenTannoudji NCKU 23 March

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. 2

PRODUCING AND MANIPULATING ULTRACOLD ATOMS AND MOLECULES • Radiative forces • Cooling • Trapping • Feshbach resonances 3

Forces exerted by light on atoms A simple example Target C bombarded by projectiles p coming all along the same direction 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 4

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 / t. R)= 10 -2 x 108 m/s 2 = 106 m/s 2 = 105 g Huge radiation pressure force! Stopping an atomic beam Laser beam Atomic beam Tapered solenoid Zeeman slower J. Prodan W. Phillips H. Metcalf 5

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 n. L (n. L < n. A) n. L < n. A 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” 6

“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 m. K range 7

Evaporative cooling H. Hess, J. M. Doyle MIT E 4 E 2 E 1 U 0 E 3 Atoms trapped in a potential well with a finite depth U 0 2 atoms with energies E 1 et E 2 undergo an elastic collision After the collision, the 2 atoms have energies E 3 et E 4, with E 1+ E 2= E 3+ E 4 If E 4 > U 0, the atom with energy E 4 leaves the well The remaining atom has a much lower energy E 3. After rethermalization of the atoms remaining trapped, the temperature decreases 8

Temperature scale (in Kelvin units) cosmic microwave background radiation (remnant of the big bang) The coldest matter in the universe 9

Traps for neutral atoms “Optical Tweezers” A. Ashkin, S. Chu Spatial gradients of laser intensity Focused laser beam. Red detuning (w. L < w. A) The light shift d. 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 “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”) 10

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 (1 D, 2 D, 3 D) and the symmetry (triangular lattice, cubic lattice) Furthermore, possibility to control atom-atom interactions, both in magnitude and sign, by using “Feshbach resonances” 11

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 12

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 hn 0 above the initial atomic state and then reemitted. There is a resonance in the scattering amplitude when n is close to n 0 13

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 them can be varied by sweeping a magnetic field V Closed channel E 0 r Open channel 14

Scattering length versus magnetic field a a> 0 Repulsive effective long range interactions a= 0 No interactions Perfect gas B abg Background scattering length a< 0 Attractive effective long range interactions Near B=B 0, Ia. I is very large Strong interactions Strong correlations B 0 : 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) 15

Bound state of the two-channel Hamiltonian In the region a » range r 0 of atom–atom interactions Eb a = a< 0 No bound state a> 0 Bound state with an energy Eb= - ħ 2 / ma 2 - (B – B 0)2 B The bound state exists only in the region a > 0. It has a spatial extension a and an energy Eb= - ħ 2 / ma 2 Weakly bound dimer with universal properties Quantum “halo” state or “Feshbach molecule” 16

Formation of a Fehbach molecule Eb a< 0 No bound state a> 0 Bound state with an energy Eb= - ħ 2 / ma 2 - B 2 B If B 0 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) 17

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 18

TESTING FUNDAMENTAL LAWS WITH ULTRACOLD ATOMS Ultraprecise Atomic Clocks 19

Measuring time with atomic clocks Principle of an atomic clock The correction loop locks the frequency of the oscillator to the frequency w. A of the hyperfine transition of 133 Cs 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 w. A. Dw 1/T Oscillator w 0 T : Observation time It is therefore interesting to use slow atoms in order to increase T, and thus to decrease D w 20

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 21

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 22

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) 23

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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 25

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) 26

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 w. A(z+dz) different from the frequency, w. 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 tests Another possible application : determination of the “geoid”, surface where the gravitational potential has a given value 27

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” 30

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 87 Rb 1995 Many others atoms have been condensed 7 Li, 1 H, 4 He*, 41 K, 133 Cs, 174 Yb, 52 Cr… MIT 23 Na 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 34

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 35

BKT crossover in a trapped 2 D atomic gas J. Dalibard group, ENS, Paris, Nature, 441, 1053 (2006) How to prepare the 2 D 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 36

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) 37

Non interacting fermions (single band model) Compression Compressible “metal” Band insulator Repulsive interactions Compression Cloud size (compressibility) Compressible “metal” Mott insulator MI BI

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 (k. BT « a) or Antiferromagnetic order in a square lattice Triangular lattice ? Frustration 39

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 40

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