Lecture II

Non dissipative trapsEvaporative cooling

Bose-Einstein condensation

Phase space density

From room temperature to 100 K

n = 10-7

Magneto-optical trap

Molasses

100 K 10 K

Intrisically limited because of the dissipative character of the MOT.

No light, no heating due to absorption

Relies on magnetic moment interaction

The force results from the inhomogenity of the magnetic field

Magnetic trapping (1)

For an atom with an nuclear spin in the ground state

F=1,m=1

F=1,m=0

F=1,m=-1

z

Maxwell's equations:No max of |B| in the vaccum.

Atoms cannot be magnetically trapped in the lower energy state.

Two-body inelastic collisions

Three-body inelastic collisions (dimer Rb2).

Ultra High Vacuum chamber, backgound gas collisions.

Photo: Bell Labs

Local minimum of |B|

+ spin polarisationV=|||B|

Non dissipative trap !!!

Magnetic trapping (2)

Spin flipsMajorana losses

Magnetic trap: classical picture versus the quantum one

Classically, the angle θ between the magnetic moment and the magnetic field is constant due to the rapid precession of µ around the magnetic field axis.

Classical picture

V=|||B|

Quantum picture

can take onlyquantized values

F=2

F=1

Magnetic trap with coils

What kind of gradient do we need ?

Gradient scales as I/d2

Magneto-optical trap: 1 mm, T=50 K

b' r = kB T b' = 10 Gauss/cm

Atoms are further compressed b' ~ 200 Gauss/cm

Two kind of solutions I ~ 1000 A, d ~ cm

I ~ 0.1 A, d ~ 100 mmMicrochip

Magnetic trap with coils

One coil:

BB0-2b'xb'yb'z

x

y

z

Two coils (antiHelmoltz):

x

y

zOB

-4b'x2b'y2b'z

B2 =4b'2 (4x2+y2+z2)

Time averaged Orbital Potential (TOP)

B-4b'x2b'y2b'z

B0B0cos(t)B0cos(t)

Quadupolar configuration

O

z

xy

Rotating field

+

=trap Larmor

100 Hz 5kHz 1 MHz

Ioffe pritchard trap

depth: 1 mK

constant bias field

gradient curvature

Microchip traps

Ioffe Pritchard traps of various aspect ratios:

Y-shaped splitting and recombining regions.

interferometry device

Atomic conveyer belt

Magnetic guide with wires

Magnetic guide with 4 tubes

2D Quadrupolar configuration

x

y

Bb'x-b'y

Add a longitudinal bias field to avoid spin flips

Evaporation

F=1,m=1

F=1,m=0

F=1,m=-1

z

radio frequency wave

Relies on the redistribution of energy through elastic collisions

Surface Evaporation

J. Low Temp. Phys. 133, 229 (2003)

works with silicon surface

Interactions between cold atoms

Two-body problem:

2 21 2

1 22 2

p pW r r

m m

One-body scattering problem

2

2

pH W r

Scattering state (eigenstate of H with a positive energy)

scattering amplitude

( ) ( , , ')ikr

ik rk

er e f k n n

r

n 'n

At low energy, and if W decreases faster than r-3 at infinity:

scattering length

Two interaction potentials with the same scattering length lead to the same properties at sufficiently low temperature

0( , , ') kf k n n a

Exceptions: dipole-dipole interactions (magnetic or electric) 1/r interactions induced by laser (Kurizki et al)

Interactions between cold atoms

Characteristic length1/ 4

62

2c

Ca

from 0.1 to 10 nmfrom 0.1 to 10 nm

W(r)W(r)

rr

66 /C r

varies rapidly with all parameters: = number of bound states

1 tanca a

75% 25% empirical law75% 25% empirical law

scat

teri

ng le

ngth

scat

teri

ng le

ngth

00 0.020.02 0.040.04-0.04-0.04 -0.02-0.026

6

C

C

a = 5 nm for 87Rb

Evaporation: a simple model (1)

1)

2)

3)

Infinite depth

Finite depth

Infinite depth

harmonic confinement

Evaporation: a simple model (2)

We deduce a power law dependence with

The phase space density changes according to

with and

N: 109 106

T: 100 K 100 nKnx 106Typical numbers

The real form of the potential only changes the exponent

Signature of condensation: time of flight

3 106 atoms in an anisotropicmagnetic trap

100 m * 5 m

0,5 to 1 K

Time of flight

T > Tc

Boltzmanngas

21 1

2 2imv kT

T < Tc

condensate

21 1

2 4i imv

isotropic expansion anisotropic expansion

CameraCCD

atomsLaser

2001 Physcis Nobel Prize E. Cornell, W. Ketterle and C. Wieman

"for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates"

Bose Einstein condensation

Review of Modern Physics, 74, 875 (2002); ibid 74, 1131 (2002)

Dipole trap gallery

Single atom in a dipole trap

possible application in quantum computing ?

Is it possible to realize a continuous source of degenerate atoms ?

PRL 93, 093003 (2004)

10 elastic collisions per atom

First signal of evaporation andgain in phasespace density