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![Page 1: Groupe de Physique des Atomes Refroidis Daniel Hennequin Olivier Houde Optical lattices Philippe Verkerk Laboratoire de Physique des Lasers, Atomes et.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649ea15503460f94ba504d/html5/thumbnails/1.jpg)
Groupe de Physique des Atomes Refroidis
Daniel HennequinOlivier Houde
Optical lattices
Philippe Verkerk
Laboratoire de Physique des Lasers, Atomes et MoléculesUniversité de Lille 1 ; Villeneuve d’Ascq ; France
A lot of work done in the former group of Gilbert Grynberg at ENS.
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Reactive Force (dipole force)
Intensity
= L - 0
Standing wave
> 0 : « blue » detuning
I, U
z
U : optical potential (light shifts)
Optical Lattices
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Outlook
I. Dissipative optical lattices1D2D3Dmore D
II. Non dissipative optical lattices
III. Instabilities in a MOT
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1D Dissipative Optical Lattice
The original one : Sisyphus cooling
J=1/2
J=3/21/3
1
- +z
E y
Ex
-1.4
-1.2
-1.0
-0.8
-0.6
-2 -1 0 1 2
= L -
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Sisyphus Cooling
-1.4
-1.2
-1.0
-0.8
-0.6
-2 -1 0 1 2
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Quantum Picture
-1.4
-1.2
-1.0
-0.8
-0.6
-2 -1 0 1 2
= 2 √ E Uvib R 0 Pump-Probe spectroscopy
Y. Castin & J. DalibardEuroPhys. Lett.
14, 761 (1991)
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Two-photon transition
Seems very difficult, but if , it is equivalent to a 1-photon transition, with : a frequency = L - p
an effective Rabi frequencyeff = p /
Two-level system :
|g>
|e>
0 Lv
|n>
|n+1>
Lorentzian
n n+1
(-v)2+n n+12
eg
(L-0)2+eg2
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Raman transitions
Position √ I / Compatible withv
Width : << ’= s’/2≈ 500 kHz/2 ≈ 50 kHz
Atomic observables not destroyed by spontaneous emission.
Lamb-Dicke effect : Raman coherences survive. n n+1 = ( n n + n+1 n+1 )/2 n n = (2n+1) ’
where = 2 ER / h v = 2 R/v
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Lamb Dicke EffectTo 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 v )1/2
First order couples | n > only to | n+1 > and | n-1>Probability to go from | n > to | n+1 > : (n+1) R/v
Probability to go from | n > to | n -1 > : n R/v
Probability to leave | n > : (2n+1) R/v
Average on ksp < | kL-ksp |2 > = 2 kL2
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Discussion
The atom scatters a lot of photons.But the momentum of a photon is small compared tothe width of the momentum distribution of the atomic state.
The momentum distribution is not changedso much in a single event.The overlap of the modified distributionwith the original one is large :
1 - (2n+1) R/v
We are far in the Lamb-Dicke regime as : R/2 = 2 kHz and v/2 ≈ 100 kHz
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Spectral analysis of the fluorescence
Spontaneous Raman transitions
Spontaneousred photon
The temperature can be deduced fromthe ratio of the 2 side-bands.But one has to be careful, because of theoptical thickness of the medium :the spontaneous photon acts as a probe for stimulated Raman transitions.
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Recoil Induced Resonance
Centered in =0Still narrower Strange shape
Nothing to dowith the lattice !
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Raman trans. in momentum space
E=px2/2m
px
Free atoms ; momentum kick : px = h k
Initial state : px, Ei=px2/2m
Final state : px+px, Ef=(px +px )2/2m
Absorption : [(px+px) - (px)]
(- Ef + Ei)2 + 2
Assuming px « <px>, and small enough
ddpx px = m / k
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Classical picture
Pump-probe interference pattern :very shallow potential moving at vx = / k
Atoms slow down while climbing hills, and accelerate coming down.As the potential is very shallow, only atoms with a velocityclose to vx = / k 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.
For zero frequency components, the pump and the probe inducea density grating. The pump diffracts on that grating, and thediffracted wave interferes with the probe gain or attenuation
The signal for is given by d (the small param. is the potential depth). dpx px = m / k
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From 1D to 2D
1D : a pair of contra-propagating waves2D : two pairs of contra-propagating wavesBad Idea !
Phase dependent potential 2 orthogonal standing-waves
-2
-1
0
1
2
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
In phase In quadrature
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Better idea
Use just 3 waveswith 120°
Linear polarization out of plane
-2
-1
0
1
2
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
Linear polarization in the plane
mg=1/2 mg=-1/2
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A few words about crystallography
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 = 2pj 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.
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3Dz
E y Ex
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And then…
1D : 2 beams2D : 3 beams3D : 4 beams4D : 5 beams…
Where is thefourth dimension ?
Consider a 3D restriction of a 4D periodic optical potential
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1D cut of a 2D potential
1
1
3
3
2
2
A 2D square lattice, but the atom can move only along a line.Depending on the slope of the line, one has different potentials.
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Periodic, super-periodic &quasi-periodic potential
1
3
2
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.
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Lissajous1.0
0.5
0.0
-0.5
-1.0
1.00.50.0-0.5-1.0-1.0
-0.5
0.0
0.5
1.0
1.00.50.0-0.5-1.0
-1.0
-0.5
0.0
0.5
1.0
1.00.50.0-0.5-1.0
r=fy/fx=1.5 r=25
r=√ 2
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The angle is small ≈ 10-2 rad
Super-lattices
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Fluorescence imagesWith the extra beam Without
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Shadow image
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Periodic, super-periodic &quasi-periodic potential
1
3
2
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.
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Long range order2.0
1.5
1.0
0.5
0.0
302520151050s
50
40
30
20
10
0
14121086420Hz
FFT
Similar patterns can be found in several places,but they differ slightly.
Larger patternslarger distances
U(x,y)=cos2x+cos2yy = x
V(x)=cos2x+cos2(x)2 frequencies
{
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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 !
} {
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Optical lattice with 5-fold symmetry
A 5-fold symmetry is incompatiblewith a translational invariance.i.e. you cannot cover the planewith pentagones.Penrose tilling.
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It works !One can measure :
the temperaturethe life timethe vibration freq.…
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20
80
40
0
210-1-2mm
τ=7ms
Δτ = 1100 ms
Z
Y
Spatial diffusion : method1. Load the atoms from the MOT in the lattice2. Wait τ3. Take an image
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Spatial diffusion :results0.5
0.4
0.3
0.2
<2 >
(m
m2 )
0.70.60.50.40.30.20.10.0
time (sec)
= direction périodique= plan quasipériodique
0.30
0.25
0.20
0.15
0.10
0.05
0.00
D (
mm
2 /sec
)
120100806040
'0 / r
= -15, = -20 (direction périodique)= -15, = -20 (plan quasipériodique)
Anisotropy in the diffusionby a factor of 2.
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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 configurationwill not trap atoms. The total field is 0 along lines.
3D trap with two beams.
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1D array of ring-shaped traps.2 contrapropagating beams with different transverse shapes,and blue detuning :
r0
I Hollow beam
Gaussian beam
r0 : possible destructive interference
r0
U
z
U/2
r0
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A conical lens
r
Intensité Expérience
Simulation
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The hollow beam
CC
D
LensMask
Telescope
Fluorescence of the hot atoms with the hollow
beam at resonance
Ring diameter : 200 µmRing width : 10 µm
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The preliminary results
Image of the atoms that remain in the lattice 80 ms after the end of the molasses. = 2 20 GHz.
Fraction (%) of the atoms that remain in the lattice vs time.
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Instabilities in a MOT
I
I
1
3
2
miroir
miroir
miroir
cellule de césium
MOT with retroreflected beams
When the laser approches the resonance, some instabilities appear both on the shape and the position of the cloud.
I will not consider here the instabilities and other rotating MOTs due to a misalignment of the beams
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The shadow effectThe 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 takeninto 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.
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The results
00
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0
ZN
t (s)
-1.0
-0.8
-0.6
-0.4
-0.2
% p
os. x
2000150010005000ms
40x106
35
30
25
# atomes
Theory
Experiment