Lecture II Non dissipative traps Evaporative cooling Bose-Einstein condensation.
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Transcript of Lecture II Non dissipative traps Evaporative 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
Microchip traps
Ioffe Pritchard traps of various aspect ratios:
Y-shaped splitting and recombining regions.
interferometry device
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
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)