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Cold atoms
Lecture 3.18th October, 2006
Non-interacting bosons in a trap
3
Useful digression: energy units
-1
B B
B
-B
B
1
/
1K 1eV s
1K / /
1eV / /
s / /
J
J
J
energy
k e k h
e k e h
h k h
e/
e
k
h/
-1
05 10
04
23
19 14
-1 31 15 41
1K 1eV s
1K 8.63 10 2.08 10
1eV 1.16 10 2.41 1
1.38 10
0
s
energy
4.80 10 4.14 1
1.60 10
6.63 100
4
Trap potential
Typical profile
coordinate/ microns
?
evaporation cooling
This is just one direction
Presently, the traps are mostly 3D
The trap is clearly from the real world, the atomic cloud is visible almost by a naked eye
5
Trap potential
Parabolic approximation
in general, an anisotropic harmonic oscillator usually with axial symmetry
2 2 2 2 2 2 21 1 1 1
2 2 2 2x y z
x y z
H m x m y m zm
H H H
p
1D
2D
3D
6
Ground state orbital and the trap potential
level number
200 nK
400 nK
0/ xx a
2
20
0 0 0 0
2 22
0 0 0 2 20 00
( , , )
1 1 1 1( ) e , ,
2 2 2
x y z
u
a
m
x y z x y z
u a Em ma Mu aa
22 2
0
1 1( )
2 2
uV u m u
a
• characteristic energy
• characteristic length
7
Ground state orbital and the trap potential
level number
200 nK
400 nK
0/ xx a
2
20
0 0 0 0
2 22
0 0 0 2 20 00
( , , )
1 1 1 1( ) e , ,
2 2 2
x y z
u
a
m
x y z x y z
u a Em ma Mu aa
22 2
0
1 1( )
2 2
uV u m u
a
0
6
87
1 m
=10 nK
~ 10 at.
Rba
N
• characteristic energy
• characteristic length
8
Ground state orbital and the trap potential
level number
200 nK
400 nK
0/ xx a
2
20
0 0 0 0
2 22
0 0 0 2 20 00
( , , )
1 1 1 1( ) e , ,
2 2 2
x y z
u
a
m
x y z x y z
u a Em ma Mu aa
22 2
0
1 1( )
2 2
uV u m u
a
0
6
87
1 m
=10 nK
~ 10 at.
Rba
N
• characteristic energy
• characteristic length
9
Filling the trap with particles: IDOS, DOS1D
2D
1
( ) int( / ) /
( ) '( )
E E E
E E
D
const.E x yE E E
212( ) /( )
( ) '( ) /( )
x y
x y
E E
E E E
D
"thermodynamic limit"
only approximate … finite systems
better for small
meaning wide trap potentials
For the finite trap, unlike in the extended gas, is not divided by volume !!( )ED
x
y
x E
10
Filling the trap with particles3D
316
212
( ) /( )
( ) '( ) /( )
x y z
x y z
E E
E E E
D
Estimate for the transition temperature
particle number comparable with the number of states in the thermal shell
B
1 1
2 2B
1 1
3 3B
2D / ( )
3D / ( )
c x y
c x y z
N k T
T k N
T k N
For 106 particles,2
B 10ck T
• characteristic energy
11
Filling the trap with particles3D
316
212
( ) /( )
( ) '( ) /( )
x y z
x y z
E E
E E E
D
Estimate for the transition temperature
particle number comparable with the number of states in the thermal shell
B
1 1
2 2B
1 1
3 3B
2D / ( )
3D / ( )
c x y
c x y z
N k T
T k N
T k N
For 106 particles,2
B important for therm. limit10 ck T
• characteristic energy
12
The general expressions are the same like for the homogeneous gas.
Working with discrete levels, we have
and this can be used for numerics without exceptions.
In the approximate thermodynamic limit, the old equation holds, only the volume V does not enter as a factor:
In 3D,
Exact expressions for critical temperature etc.
0( )
1( , )
e 1VTN
N
( )0
1d ( )
e 1
D
( )
1( , ) ( )
e 1jjj j
nN T
N
1 1 1
3 3 3B B( (3)) / 0.94 /cT k N k N
3BE 1 ( / ) ,c cN N T T T T
for0 CT T
13
How good is the thermodynamic limit1D illustration (almost doable)
0( ) )(
1 11 1d
e 1 e 1 e 1jjN
?
14
How good is the thermodynamic limit1D illustration
0( ) )(
1 11 1d
e 1 e 1 e 1jjN
?
15Bk T
15
How good is the thermodynamic limit1D illustration
0( ) )(
1 11 1d
e 1 e 1 e 1jjN
?
/
red bars… the sum
firstterm
continuousBE step function,
whose integralequals the rest
of the sum
15Bk T
16
How good is the thermodynamic limit1D illustration
0( ) )(
1 11 1d
e 1 e 1 e 1jjN
?
/
red bars… the sum
firstterm
continuousBE step function,
whose integralequals the rest
of the sum
15Bk T
The quantitative criterion for the thermodynamic limit
1B Ck T
17
How sharp is the transition
These are experimental data
fitted by the formula
The rounding is apparent,
but not really an essential feature
3BE 1 ( / ) ,c cN N T T T T
18
Seeing the condensate – reminescence of L2
double average, quantum and thermal
= insert unit operator
chang
de
e the summation
fine the one-particle density mat
rix
order
|
Tr
X n
X n
n X
X
X
X
= n
Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix.
Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:
19
OPDM in the Trap
• Use the eigenstates of the 3D oscillator
• Use the BE occupation numbers
• Single out the ground state
( )
= ( , , ), 0, 1, 2, 3,
1
= , = e 1
= = + + x y z
x y z w
x y z
x x x x x x
E
n
E E E E
(000)
BEC TERM
( )
1 1= 000 000
e 1 e 1
E
zero point oscillations
absorbed in the
chemical potential
20
OPDM in the Trap
• Use the eigenstates of the 3D oscillator
• Use the BE occupation numbers
• Single out the ground state
( )
= ( , , ), 0, 1, 2, 3,
1
= , = e 1
= = + + x y z
x y z w
x y z
x x x x x x
E
n
E E E E
(000)
BEC TERM
( )
1 1= 000 000
e 1 e 1
E
zero point oscillations
absorbed in the
chemical potentialCoherent component, be it condensate or not. At , it contains
ALL atoms in the cloud/ BT k
Incoherent thermal component, coexisting with the
condensate. At , it freezes out and contains
NO atoms
/ BT k
21
OPDM in the Trap, Particle Density in Space
The spatial distribution of atoms in the trap is inhomogeneous. Proceed by definition:
BEC THERM( )
n r r r r r r r
as we would write down
naively at once
op
2
( )
( )
( ) Tr ( )
Tr d ( ) Tr
1
e 1
1( )
e 1
E
E
n
r r r
r r r r r r r
r r r r
r
Split into the two parts, the coherent and the incoherent phase
22
OPDM in the Trap, Particle Density in Space
THERM
BEC THERM
(000)
22
000(000)
BEC
( )
( )
known laborious
( )
1 1= 000 000
e 1 e 1
1 1( ) ( )
e 1 e 1
( ) ( )
E
E
n
n n
r r r r r r r
r r r r
r r
r r
Split into the two parts, the coherent and the incoherent phase
23
OPDM in the Trap, Particle Density in Space
THERM
BEC THERM
(000)
22
000(000)
BEC
( )
( )
known laborious
( )
1 1= 000 000
e 1 e 1
1 1( ) ( )
e 1 e 1
( ) ( )
E
E
n
n n
r r r r r r r
r r r r
r r
r r
Split into the two parts, the coherent and the incoherent phase
22 2
2 2 20 0 0
22 2
BEC 0 0 0
30 0 0
1( )
e 1
1 1e
e 1x y z
x y z
x y z
yx za a a
n x y z
a a a
rThe characteristic lengths directly observable
24
Particle Density in Space: Boltzmann Limit
We approximate the thermal distribution by its classical limit.
Boltzmann distribution in an external field:
2 2 2 2 2 2
3THERM
( ( ))
( )
1 )2
(
( , ) e
( ) d ( , )
e
ex y z
B
B
W U
U
x y zm
f
n f
r
r
r p
r p r p
25
Particle Density in Space: Boltzmann Limit
We approximate the thermal distribution by its classical limit.
Boltzmann distribution in an external field:
2 2 2 2 2 2
3THERM
( ( ))
( )
1 )2
(
( , ) e
( ) d ( , )
e
ex y z
B
B
W U
U
x y zm
f
n f
r
r
r p
r p r p
For comparison:
22 2
2 2 20 0 0
22 2
BEC 0 0 0
30 0 0
1( )
e 1
1 1e
e 1x y z
x y z
x y z
yx za a a
n x y z
a a a
r
26
Particle Density in Space: Boltzmann Limit
We approximate the thermal distribution by its classical limit.
Boltzmann distribution in an external field: Two directly observable characteristic lengths
2 2 2 2 2 2
3THERM
( ( ))
( )
1 )2
(
( , ) e
( ) d ( , )
e
ex y z
B
B
W U
U
x y zm
f
n f
r
r
r p
r p r p
For comparison:
1
30 0
2
0
0
0
0 ,
1
(
/
)x y z
B
R
m
m
a
a
T
a a
k T
a
a
1
3( )x y z
22 2
2 2 20 0 0
22 2
BEC 0 0 0
30 0 0
1( )
e 1
1 1e
e 1x y z
x y z
x y z
yx za a a
n x y z
a a a
r
27
Particle Density in Space: Boltzmann Limit
We approximate the thermal distribution by its classical limit.
Boltzmann distribution in an external field: Two directly observable characteristic lengths
2 2 2 2 2 2
3THERM
( ( ))
( )
1 )2
(
( , ) e
( ) d ( , )
e
ex y z
B
B
W U
U
x y zm
f
n f
r
r
r p
r p r p
For comparison:
1
30 0
2
0
0
0
0 ,
1
(
/
)x y z
B
R
m
m
a
a
T
a a
k T
a
a
1
3( )x y z
anisotropygiven by analogous
definitions of thetwo lengths
for each direction
22 2
2 2 20 0 0
22 2
BEC 0 0 0
30 0 0
1( )
e 1
1 1e
e 1x y z
x y z
x y z
yx za a a
n x y z
a a a
r
28
Real space Image of an Atomic Cloud
29
Real space Image of an Atomic Cloud
30
Real space Image of an Atomic Cloud
31
Real space Image of an Atomic Cloud
• the cloud is macroscopic
• basically, we see the thermal distribution
• a cigar shape: prolate rotational ellipsoid
• diffuse contours: Maxwell – Boltzmann distribution in a parabolic potential
32
Particle Velocity (Momentum) DistributionThe procedure is similar, do it quickly:
THERM
BEC THERM
(000)
2 2
000(000)
BEC
( )
( )
known
laborious
( )
1 1= 000 000
e 1 e 1
1
1( ) ( )
e 1 e
1
( ) ( )
E
E
f
f f
p p p p p p p
p p p p
p p
r r
22 2
2 2 20 0 0
2
00
2 2
BEC 0 0 0
1( )
e 1
1e ,
e 1
yx z
x y z
x x y y z z
ww
pp pb b b
f p p p
ba
p
33
Thermal Particle Velocity (Momentum) DistributionAgain, we approximate the thermal distribution by its classical limit.
Boltzmann distribution in an external field:
1 2 2 2
3THERM
( ( ))
1 )2
(
( , ) e
( ) d ( , )
e
ex y z
B
B
W U
W
m p p p
f
f f
rr p
r r r p
22 2
2 2 20 0 0
2
00
2 2
BEC 0 0 0
1( )
e 1
1e ,
e 1
yx z
x y z
x x y y z z
ww
pp pb b b
f p p p
ba
p
Two directly observable characteristic lengths
00
00
1
30 0 0 ,
1
(
/
)x y z
B
ba
b b b
T
k T
m
b
B
b
Remarkable:
Thermal and condensatelengths in the same ratio
for positions and momenta
0 0
0 0
/
/
B
B
B b k T bT
R a k T aT
34
BEC observed by TOF in the velocity distribution
Qualitative features: all Gaussians
wide vz.narrow
isotropic vs. anisotropic
The end
36
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