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.

Reactive Force (dipole force)

Intensity

= L - 0

Standing wave

> 0 : « blue » detuning

I, U

z

U : optical potential (light shifts)

Optical Lattices

Outlook

I. Dissipative optical lattices1D2D3Dmore D

II. Non dissipative optical lattices

III. Instabilities in a MOT

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 -

Sisyphus Cooling

-1.4

-1.2

-1.0

-0.8

-0.6

-2 -1 0 1 2

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)

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

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

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

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

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.

Recoil Induced Resonance

Centered in =0Still narrower Strange shape

Nothing to dowith the lattice !

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

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

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

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

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.

3Dz

E y Ex

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

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.

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.

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

The angle is small ≈ 10-2 rad

Super-lattices

Fluorescence imagesWith the extra beam Without

Shadow image

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.

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

{

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 incompatiblewith a translational invariance.i.e. you cannot cover the planewith pentagones.Penrose tilling.

It works !One can measure :

the temperaturethe life timethe vibration freq.…

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

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.

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.

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

A conical lens

r

Intensité Expérience

Simulation

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

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.

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

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.

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