Groupe de Physique des Atomes Refroidis

1. Laboratoire de Physique des Lasers, Atomes et MoléculesUniversité de Lille 1 ; Villeneuve d’Ascq ; France Groupe de Physique des Atomes Refroidis Optical lattices Philippe Verkerk Daniel Hennequin Olivier Houde A lot of work done in the former group of Gilbert Grynberg at ENS.

2. Optical Lattices Reactive Force (dipole force) Intensity  = L - 0 I, U z U : optical potential (light shifts) Standing wave  > 0 : « blue » detuning

3. Outlook I. Dissipative optical lattices 1D 2D 3D more D II. Non dissipative optical lattices III. Instabilities in a MOT

4. J=3/2 1/3 1 J=1/2 z - + E x E y 1D Dissipative Optical Lattice The original one : Sisyphus cooling D = wL - w0

5. Sisyphus Cooling

6. W = 2 √ E U vib R 0 Quantum Picture Y. Castin & J. Dalibard EuroPhys. Lett. 14, 761 (1991) Pump-Probe spectroscopy

7.  |e> Geg (wL-w0)2+Geg2 |n+1>   Wv w0 wL Gn n+1 (d-Wv)2+gn n+12 d  |g> |n> Two-photon transition Seems very difficult, but if D >> G , W, it is equivalent to a 1-photon transition, with :  a frequency d = wL - wp  an effective Rabi frequency Weff = W Wp / D Two-level system : Lorentzian

8. Position  √ I / D Compatible with Wv Lamb-Dicke effect : Raman coherences survive. gn n+1 = (gn n + gn+1 n+1 )/2 gn n = (2n+1) h2 G’ where h2 = 2 ER / h Wv = 2 wR/Wv Raman transitions Width : g << G’= G s G’/2p≈ 500 kHz g /2p ≈ 50 kHz Atomic observables not destroyed by spontaneous emission.

9. We have to evaluate < n | exp i(kL-ksp)R | n’ > Assume k.R = k Z is small, and expand the exp exp i( k Z ) = 1 + i k Z + … Z = ( a + a†) ( h / 2m Wv )1/2 First order couples | n > only to | n+1 > and | n-1> Probability to go from | n > to | n+1 > : (n+1) wR/Wv Probability to go from | n > to | n -1 > : n wR/Wv Probability to leave | n > : (2n+1) wR/Wv Lamb Dicke Effect To evaluate the decay rate of the population of state |n> we have to consider the recoil due to spontaneous emission. The atom, close to R=0, absorbs a photon kL and emits a photon ksp The spatial part of the coupling is : exp i(kL-ksp)R Average on ksp < | kL-ksp |2 > = 2 kL2

10. Discussion The atom scatters a lot of photons. But the momentum of a photon is small compared to the width of the momentum distribution of the atomic state. The momentum distribution is not changed so much in a single event. The overlap of the modified distribution with the original one is large : 1 - (2n+1) wR/Wv We are far in the Lamb-Dicke regime as : wR/2p = 2 kHz and Wv/2p ≈ 100 kHz

11. Spontaneous red photon Spectral analysis of the fluorescence Spontaneous Raman transitions The temperature can be deduced from the ratio of the 2 side-bands. But one has to be careful, because of the optical thickness of the medium : the spontaneous photon acts as a probe for stimulated Raman transitions.

12. Recoil Induced Resonance Centered in d=0 Still narrower Strange shape Nothing to do with the lattice !

13. Free atoms ; momentum kick : Dpx = h k q Absorption : g [P(px+Dpx) - P(px)] (d - Ef + Ei)2 + g2 Assuming Dpx« <px>, and g small enough dP dpx px = m d / k q Raman trans. in momentum space Initial state : px, Ei=px2/2m Final state : px+Dpx, Ef=(px +Dpx )2/2m E=px2/2m px

14. For zero frequency components, the pump and the probe induce a density grating. The pump diffracts on that grating, and the diffracted wave interferes with the probe gain or attenuation Pump-probe interference pattern : very shallow potential moving at vx = d / k q The signal for d is given by dP (the small param. is the potential depth). dpx px = m d / k q Classical picture Atoms slow down while climbing hills, and accelerate coming down. As the potential is very shallow, only atoms with a velocity close to vx = d / k q can feel the potential. If vx > 0, you have more atoms with v < vx than atoms with v > vx The density grating is following the interference pattern.

15. From 1D to 2D Bad Idea ! 1D : a pair of contra-propagating waves 2D : two pairs of contra-propagating waves Phase dependent potential 2 orthogonal standing-waves In phase In quadrature

16. Linear polarization out of plane Better idea Use just 3 waves with 120° Linear polarization in the plane mg=1/2 mg=-1/2

17. A few words about crystallography Etot = SEj exp-i kj.r = exp-i k1.rSEj exp-i(kj - k1).r For any translation R such that (kj - k1).R = 2pjp the field is unchanged. R : vectors of the lattice (position space) {kj - k1}j>1 : basis of the reciprocal lattice (Brillouin zone). If they are (d-1) independent vectors. In the case of 2 orthogonal standing waves, (k2 - k1)= (k3 - k1) + (k4 - k1) because k2 = - k1 and k3 = - k4 The problem of phase dependence is also related to that. With 3 beams in 2D, one can cancel the phases by an appropriate choice of the origins in space and in time.

18. z E x E y 3D

19. And then… 1D : 2 beams 2D : 3 beams 3D : 4 beams 4D : 5 beams… Where is the fourth dimension ? Consider a 3D restriction of a 4D periodic optical potential

20. 2 3 1 1 2 3 1D cut of a 2D potential A 2D square lattice, but the atom can move only along a line. Depending on the slope of the line, one has different potentials.

21. The slope is a simple rational number : Periodic potential The slope is a large integer : Super-periodic The slope is not a rational number : Quasi-periodic. 1 2 3 Periodic, super-periodic &quasi-periodic potential

22. r=√ 2 Lissajous r=fy/fx=1.5 r=25

23. Super-lattices The angle q is small q ≈ 10-2 rad

24. Fluorescence images With the extra beam Without

25. Shadow image

26. The slope is a simple rational number : Periodic potential The slope is a large integer : Super-periodic The slope is not a rational number : Quasi-periodic. 1 2 3 Periodic, super-periodic &quasi-periodic potential In a quasi-periodic potential, the invariance by translation is lost. But a long range order remains.

27. { U(x,y)=cos2x+cos2y y = a x V(x)=cos2x+cos2(ax) 2 frequencies FFT Larger patterns larger distances Long range order Similar patterns can be found in several places, but they differ slightly.

28. The conductivity is related to the mobility of the electrons in the potential of the ionic lattice. Ionic potential Optical potential Electrons Atoms Study the diffusion of atoms in a quasi-periodic potential ! } { Toy model for solid state physics Quasi-crystals with five-fold symmetry have been found in 1984. An alloy formed with Al, Pd and Mn, which are 3 metals (with a good conductivity), is almost an insulator (8 orders of magnitude). What is the role of the quasi-periodicity ?

29. Optical lattice with 5-fold symmetry A 5-fold symmetry is incompatible with a translational invariance. i.e. you cannot cover the plane with pentagones. Penrose tilling.

30. It works ! One can measure : the temperature the life time the vibration freq. …

31. Spatial diffusion : method 1. Load the atoms from the MOT in the lattice 2. Wait Dt 3. Take an image

32. = direction périodique = plan quasipériodique 0.5 ) 2 0.4 > (mm 2 s 0.3 < 0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 time (sec) 0.30 G G = -15 , = -20 (direction périodique) 0.25 G G = -15 , = -20 (plan quasipériodique) 0.20 /sec) 2 0.15 D (mm 0.10 0.05 0.00 40 60 80 100 120 D w ' / 0 r Spatial diffusion :results Anisotropy in the diffusion by a factor of 2.

33. Far detuned lattices Red detuning : it works nicely ! but the atoms see a lot of light. Blue detuning : the atoms are in the dark ! for the same depth, less scattered photons Be careful in the design : the standard 4 beams configuration will not trap atoms. The total field is 0 along lines. 3D trap with two beams.

34. I Hollow beam Gaussian beam r 0 r0 : possible destructive interference r0 U U /2 r z 0 1D array of ring-shaped traps. 2 contrapropagating beams with different transverse shapes, and blue detuning :

35. Intensité Expérience Simulation r A conical lens

36. Telescope CCD Mask Lens The hollow beam Fluorescence of the hot atoms with the hollow beam at resonance Ring diameter : 200 µm Ring width : 10 µm

37. The preliminary results Image of the atoms that remain in the lattice 80 ms after the end of the molasses. D = 2p 20 GHz. Fraction (%) of the atoms that remain in the lattice vs time.

38. 1 cellule de césium s+ s+ miroir I s+ 2 miroir s- I 3 s- s- miroir Instabilities in a MOT MOT with retroreflected beams I will not consider here the instabilities and other rotating MOTs due to a misalignment of the beams When the laser approches the resonance, some instabilities appear both on the shape and the position of the cloud.

39. The shadow effect The beams are retro-reflected. The cloud of cold atoms absorbs part of the power. The backward beam is weaker than the forward one. The cloud is then pushed away from the center. We measure the displacement with a segmented photodiode. We can consider a 1D system with only global variables :  the number of atoms in the cloud, N  the motion of its center of mass, z and v. The repulsion due to multiple scattering has not to be taken into account, because it is an internal force. Assuming that the efficiency of the trapping process depends on the position of the center of mass, we obtain a set of three non-linear coupled equations. Numerical solutions.

40. -1.0 6 40x10 -0.8 % pos. x 35 -0.6 # atomes 30 -0.4 -0.2 25 0 500 1000 1500 2000 ms The results N Z Theory t (s) Experiment