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Laboratoire de Physique
des Lasers, Atomes et Molécules
Université 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.
Optical Lattices
Reactive Force (dipole force)
Intensity
U : optical potential
(light shifts)
 = L - 0
I, U
Standing wave
 > 0 : « blue » detuning
z
Outlook
I. Dissipative optical lattices
1D
2D
3D
more D
II. Non dissipative optical lattices
III. Instabilities in a MOT
1D Dissipative Optical Lattice
J=3/2
1/3
The original one : Sisyphus cooling
Ex
z
-
1
J=1/2
+
 = L - 0
Ey
-0.6
-0.8
-1.0
-1.2
-1.4
-2
-1
0
1
2
Sisyphus Cooling
-0.6
-0.8
-1.0
-1.2
-1.4
-2
-1
0
1
2
Quantum Picture
Y. Castin & J. Dalibard
EuroPhys. Lett.
14, 761 (1991)
-0.6
-0.8
-1.0
-1.2
-1.4
-2
W vib = 2 √ E R U0
-1
0
1
2
Pump-Probe spectroscopy
Two-photon transition
Seems very difficult, but if  >> G , W,
it is equivalent to a 1-photon transition,
with :  a frequency d = L - p
 an effective Rabi frequency
Weff = W Wp / 
Two-level system :

|e> |n+1>
 
 d
Wv 0
L

|g> |n>
Lorentzian
Geg
(L-0)2+Geg2
Gn n+1
(d-Wv)2+gn n+12
Raman transitions
Position  √ I / 
Compatible with Wv
Width : g << G’= G s
G’/2p≈ 500 kHz
g /2p ≈ 50 kHz
Atomic observables not destroyed by spontaneous emission.
Lamb-Dicke effect : Raman coherences survive.
g n n+1 = (g n n + g n+1 n+1 )/2
g n n = (2n+1) h2 G’
where h2 = 2 ER / h Wv = 2 R/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
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) R/Wv
Probability to go from | n > to | n -1 > : n R/Wv
Probability to leave | n > : (2n+1) R/Wv
Average on ksp
< | kL-ksp |2 > = 2 kL2
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) R/Wv
We are far in the Lamb-Dicke regime as :
R/2p = 2 kHz and Wv/2p ≈ 100 kHz
Spectral analysis of the fluorescence
Spontaneous Raman transitions
Spontaneous
red photon
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.
Recoil Induced Resonance
Centered in d=0
Still narrower
Strange shape
Nothing to do
with the lattice !
Raman trans. in momentum space
Free atoms ; momentum kick : px = h k q
Initial state : px, Ei=px2/2m
Final state : px+px, Ef=(px +px )2/2m
E=px2/2m
Absorption : g [P(px+px) - P(px)]
(d - Ef + Ei)2 + g2
px
Assuming px « <px>,
and g small enough
dP
dpx px = m d / k q
Classical picture
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
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.
The signal for d is given by
(the small param. is the potential depth).
dP
dpx px = m d / k q
From 1D to 2D
1D : a pair of contra-propagating waves
Bad
2D : two pairs of contra-propagating waves
Phase dependent potential
2 orthogonal standing-waves
In phase
In quadrature
2
2
1
1
0
0
-1
-1
-2
-2
-2
-1
0
Idea !
1
2
-2
-1
0
1
2
Better idea
Linear polarization
out of plane
2
1
Use just 3 waves
with 120°
0
-1
Linear polarization in the plane
-2
-2
2
2
1
1
-1
0
1
mg=1/2
0
0
-1
-1
-2
-2
-2
-1
0
1
2
2
mg=-1/2
-2
-1
0
1
2
A few words about crystallography
S
S
Etot =
Ej exp-i kj.r = exp-i k1.r
Ej 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.
Ey
3D
Ex
z
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
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.
2
3
1
1
2
3
Periodic, super-periodic &
quasi-periodic potential
1
2
3
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.
Lissajous
1.0
0.5
1.0
0.5
0.0
0.0
-0.5
-0.5
r=√ 2
-1.0
-1.0
-0.5
0.0
0.5
1.0
1.0
-1.0
-1.0
r=fy/fx=1.5
-0.5
0.0
r=25
0.5
0.0
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
0.5
1.0
Super-lattices
The angle q is small
q ≈ 10-2 rad
Fluorescence images
With the extra beam
Without
Shadow image
Periodic, super-periodic &
quasi-periodic potential
1
2
3
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.
In a quasi-periodic potential, the invariance by translation is lost.
But a long range order remains.
Long range order
2.0
1.5
1.0
0.5
0.0
0
5
10
Similar patterns can be
found in several places,
but they differ slightly.
15
s
20
25
{
U(x,y)=cos2x+cos2y
y=ax
50
40
30
V(x)=cos2x+cos2(ax)
2 frequencies
20
Larger patterns
larger distances
30
10
0
0
2
4
6
8
Hz
10
12
14
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 ?
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 !
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.
It works !
One can measure :
the temperature
the life time
the vibration freq.
…
Spatial diffusion : method
Y
  00 m s
1. Load the atoms from
the MOT in the lattice
2. Wait 
3. Take an image
  00 m s
20
Z
80
40
0
-2
-1
0
1
2
Spatial diffusion :results
= direction périodique
= plan quasipériodique
0.4
0.3
0.30
0.2
= -15G,
= -15G,
0.25
0.1
0.2
0.3
0.4
0.5
0.6
= -20G (direction périodique)
= -20G (plan quasipériodique)
0.20
0.7
2
0.0
D (mm /sec)
2
2
< s > (mm )
0.5
time (sec)
Anisotropy in the diffusion
by a factor of 2.
0.15
0.10
0.05
0.00
40
60
80
'0 / r
100
120
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.
1D array of ring-shaped traps.
2 contrapropagating beams with different transverse shapes,
and blue detuning :
I
Hollow beam
0
U
Gaussian beam
r
r0 : possible destructive interference
U
/2
z
0
r0
r
A conical lens
Intensité
Expérience
Simulation
r
Telescope
CCD
The hollow beam
Mask
Lens
Fluorescence of the hot
atoms with the hollow
beam at resonance
Ring diameter : 200 µm
Ring width : 10 µm
The preliminary results
Image of the atoms that remain in
the lattice 80 ms after the end of
the molasses.  = 2p 20 GHz.
0.16
0.14
 = 34 ms
Ninf = 0. 07%
0.12
0.10
0.08
0.06
0.04
0.02
0.00
40
60
80
100
t (ms)
120
140
Fraction (%) of the atoms that
remain in the lattice vs time.
Instabilities
in
a
1MOT
cellule de césium
s+
MOT with retroreflected beams
s+
I will not consider here the I
instabilities and other
rotating MOTs due to a 2
misalignment of the beams
3
When the laser approches the
resonance, some instabilities
appear both on the shape and
the position of the cloud.
miroir
s+
ss-
I
s-
miroir
miroir
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.
The results
1.0
N
0.6
Z
0.8
0.4
Theory
0.2
t (s)
0
0
0.2
0.4
0.6
0.8
1.0
-1.0
-0.8
40x10
-0.6
35
-0.4
30
-0.2
25
# atomes
% pos. x
Experiment
0
500
1000
ms
1500
2000
6