Transcript Document

Lecture II
Non dissipative traps
Evaporative cooling
Bose-Einstein condensation
Phase space density
Magneto-optical trap
From room temperature to
100 mK
Molasses
100 mK
10 mK
nl3 = 10-7
Intrisically limited because
of the dissipative character of
the MOT.
Magnetic trapping (1)
No light, no heating due to absorption
Relies on magnetic moment interaction
The force results from the inhomogenity of the magnetic field
For an atom with an nuclear spin in the ground state
Magnetic trapping (2)
F=1,m=1
F=1,m=0
z
F=1,m=-1
Local minimum of |B|
Bell Labs
+Photo:
spin polarisation
V=|m||B|
Maxwell's equations:
No max of |B| in the
vaccum.
Non dissipative trap !!!
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.
Magnetic trap: classical picture versus the
quantum one
Classical picture
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.
F=2
Quantum picture
V=|m||B|
Spin flips
Majorana losses
m can take only
quantized values
F=1
mB
mB/2
0
-mB/2
-mB
mB/2
0
-mB/2
Magnetic trap with coils
What kind of gradient do we need ?
Magneto-optical trap: 1 mm, T=50 mK
m b' r = kB T
b' = 10 Gauss/cm
Atoms are further compressed
Two kind of solutions
b' ~ 200 Gauss/cm
I ~ 1000 A, d ~ cm
Gradient scales as I/d2
I ~ 0.1 A, d ~ 100 mm
Microchip
Magnetic trap with coils
One coil:
B
y
B0-2b'x
b'y
b'z
x
z
Two coils (antiHelmoltz):
y
B
-4b'x
2b'y
2b'z
O
B2 =4b'2 (4x2+y2+z2)
x
z
Time averaged Orbital Potential (TOP)
z
Quadupolar configuration
+
x
-4b'x
2b'y
2b'z
B
O
Rotating field
B
0
B0cos(wt)
B0cos(wt)
=
wtrap < < w < < wLarmor
100 Hz
5kHz
1 MHz
y
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
y
x
B
b'x
-b'y
Add a longitudinal bias
field to avoid spin flips
Evaporation
radio
frequency
wave
F=1,m=1
F=1,m=0
z
F=1,m=-1
Relies on the redistribution of energy through elastic collisions
Surface Evaporation
works with silicon surface
J. Low Temp. Phys. 133, 229 (2003)
Interactions between cold atoms
One-body scattering problem
p2
H
 W r 
2m
Two-body problem:
p12 p22

 W  r1 - r2 
2m 2m
Scattering state (eigenstate of H with a positive energy)
 k ( r ) eik r  f ( k , n , n ')
e
ikr
n'
n
r
scattering amplitude
At low energy, and if W decreases faster than r-3 at infinity:
k 0
f (k , n , n ') 
 -a
scattering length
Two interaction potentials with the same scattering length lead
to the same properties at sufficiently low temperature
Exceptions: dipole-dipole interactions (magnetic or electric)
1/r interactions induced by laser (Kurizki et al)
Interactions between cold atoms
W(r)
1/ 4
r
 2 m C6 
a

Characteristic length c  2 


-C6 / r 6
from 0.1 to 10 nm
a  ac 1 - tan F 
scattering length
F varies rapidly with all parameters: F/p = number of bound states
-0.04 -0.02
75% 25% empirical law
0
 C6
0.02 0.04 C6
a = 5 nm for 87Rb
Evaporation: a simple model (1)
1)
Infinite depth
2)
Finite depth
3)
Infinite depth
harmonic confinement
Evaporation: a simple model (2)
We deduce a power
law dependence
with
The phase space density changes according to
and
with
The real form of the potential only changes the exponent
Typical numbers
N: 109
T: 100 mK
106
100 nK
nl3 x 106
Signature of condensation: time of flight
Laser
3
106
atoms
Camera
CCD
100 mm * 5 mm
0,5 to 1 mK
atoms in an anisotropic
magnetic trap
Time of flight
T > Tc
T < Tc
Boltzmann
gas
condensate
1 2 1
mvi 
wi
2
4
1 2 1
mvi  kT
2
2
isotropic expansion
anisotropic expansion
Bose Einstein condensation
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"
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 ?
10 elastic
collisions per
atom
First signal of
evaporation and
gain in phase
space density
PRL 93, 093003 (2004)