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Institut für Theoretische Physik und Astrophysik
Universität Würzburg
Bose-Einstein Condensation
M.N.Kiselev
Experimental observation of BEC
In dilute gases of Alkali metals
Annual number of published papers, which have
the words “Bose” and “Einstein” in their title,
abstracts or keywords (ISI database)
Nobel Prize in Physics 2001
For the achievement of Bose-Einstein Condensation in dilute gases of Alkali metals…
Eric A.Cornell (USA), Wolfgang Ketterle (Germany), Carl E. Wieman (USA)
Outlook
• Symmetry
properties of many-particle
wave functions: Fermions and Bosons.
• Statistics of Bosons:
Bose-Einstein distribution function.
• Ideal Bose gas. Bose-Einstein Condensation.
• Thermodynamics of the Ideal Bose gas.
• Weakly interacting Bose gas.
• Experiments and perspectives.
*) I acknowledge the use of materials and slides from W.Ketterle Nobel Lecture 2001 and his talk
given at MIT’s Teachers Program 24/06/03. Some pictures were lent to me by courtesy of W.Ketterle.
3 particles, total energy = 3 (Arbitrary units)
3
2
1
0
Identical,
but classically distinguishable
3
2
1
0
3
2
1
0
3
6
9
12
10%
20%
30%
40%
3 particles, total energy = 3 (Arbitrary units)
3
2
1
0
3
2
1
0
3
2
1
0
3 particles, total energy = 3 (Arbitrary units)
3
2
1
0
3
2
1
0
3
2
1
0
3 particles, total energy = 3 (Arbitrary units)
3
2
1
0
3
2
1
0
3
2
1
0
Identical,
indistinguishable
3 particles, total energy = 3 (Arbitrary units)
3
2
1
0
3
2
1
0
3
2
1
0
Bosons
3 particles, total energy = 3 (Arbitrary units)
3
2
1
0
3
2
1
0
3
2
1
0
1
1
4
3
bosons
classical
11%
11%
44%
33%
10%
20%
30%
40%
Bosons
3 particles, total energy = 3 (Arbitrary units)
3
2
1
0
Classical
10 % probability
for triple occupancy
3
2
1
0
3
2
1
0
30 % probability
for double occupancy
3 particles, total energy = 3 (Arbitrary units)
3
2
1
0
Bosons
33 % probability
for triple occupancy
3
2
1
0
3
2
1
0
Bosons like to get together!
33 % probability
for double occupancy
Thermodynamics of 3-dimensional Ideal Bose Gas.
Nε =0
⎛ ⎛ T ⎞3/ 2 ⎞
= N ⎜1 − ⎜ ⎟ ⎟ ,
⎜ ⎝ Tc ⎠ ⎟
⎝
⎠
⎛T ⎞
E = 0.770 Nk BT ⎜ ⎟
⎝ Tc ⎠
5E
⎛ ∂E ⎞
CV = ⎜
,
⎟ =
⎝ ∂T ⎠V 2T
T
CV
5E
S = ∫ dT =
3T
T
0
∞
k 2 dk 2 k 2
E = ∑ ε k nk = V ∫
2
π
2
2m
k
0
3/ 2
= 0.1289
m3/ 2 (k BT )5/ 2
3
1
⎛ 2k 2
exp ⎜
⎝ 2mk BT
V,
2
F = E − TS = − E
3
m
⎛ ∂F ⎞
P = −⎜
⎟ = 0.0851
⎝ ∂V ⎠T
3/ 2
(k BT )5/ 2
Pressure does not depend on the volume
T →0
P ⎯⎯⎯
→0
3
⎞
⎟ −1
⎠
Summary:
⎛N⎞
T < Tc = 3.31
⎜ ⎟
mk B ⎝ V ⎠
2
2/3
particles start to collect at lowest energy until at T=0 they are all there
Bose Gas undergoes a phase transition without any interaction!
All the thermodynamic quantities are continuous at the transition point.
3rd - order phase transition
N0
Order parameter
λ
CV
-point in He4
Tc − T
Tc
T
Tc
T
Weakly-interacting Bose Gas
a
d
Weak interactions:
Ideal Bose Gas
Interacting
Bose Gas
ε ( p) =
a
d
p2
ε ( p) =
2m
4π 2
U (q) ≈
a
m
a – scattering length,
d – average distance
2
⎧ p
n0 a 3
p2
2
4π n 0 a ,
⎪⎪
4π n 0 a 2 2 ⎛ p 2 ⎞
2m
ma 2
m
p +⎜
⎟ ≈⎨
2
2
m2
m
n0 a 3
p2
p2
⎝
⎠
⎪
,
⎪⎩
2m
2m
ma 2
n0 =
N
V
1/ 2
N
8 3/ 2 ⎛ N ⎞
≈ 1−
a ⎜ ⎟
N0
3 π
⎝V ⎠
Particles of a non-ideal Bose gas do not
all have zero momentum,
even in the ground state.
Ordinary light
divergent
incoherent
many small waves
many modes
Laser light
diffraction limited (directional)
coherent
one big wave
single mode (monochromatic)
Ordinary gas
atoms move around randomly
divergent
incoherent
many small waves
many modes
Bose-Einstein
condensate
atoms in a coherent state
diffraction limited (directional)
coherent
one big wave
single mode (monochromatic)
Possible candidates for the experimental observation of the
Bose-Einstein Condensation
U
⎛N⎞
Tc = 3.31
⎜ ⎟
2mk B ⎝ V ⎠
2
2/3
Spin polarized hydrogen
R
Atomic Hydrogen
H2
Helium
molecular
crystal
Only 8% of particles in the condensate
Strongly interacting Bose system!
Excitons in semiconductors
electron
Strong many-body effects
hole
BEC @ JILA, June ‘95
(Rubidium)
BEC @ MIT, Sept. ‘95 (Sodium)
Distribution function
Sodium, MIT Group,
September 1995
∼ 106
particles in BEC
Tc ∼ 1µ K
field of view
Distribution function
200 µ m × 270µ m
time – 1/20 s
Rubidium, JILA Group,
June 1995
∼ 103
particles in BEC
JILA –Joint Institute for
Laboratory Astrophysics
λdB = h/pT ∝ T -1/2
BEC=Tool for knowledge
Condensed matter physics
Many-body physics
Statistical physics
• Superfluidity
• Quantum gases
• Mesoscopic physics
BEC=Tool for applications
Metrology
• Atomic clocks
• Matter wave sensors
Collisional physics
• Ultracold collisions
• Cold chemistry
Quantum optics
• Coherence of atoms
• Atom laser
• Entanglement
Visionary long-term goals
• Atom deposition – nanotechnology
• Concepts for quantum computer
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