bose-einstein condensationmkiselev/lectures/lecture02/bosons.pdf · for the achievement of...
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Bose-Einstein CondensationM.N.Kiselev
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Annual number of published papers, which have the words “Bose” and “Einstein” in their title, abstracts or keywords (ISI database)
Experimental observation of BECIn dilute gases of Alkali metals
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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)
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3 particles, total energy = 3 (Arbitrary units)
3
012
0123
12 40%9 30%6 20%3 10%
3
012
Identical, but classically distinguishable
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3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
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3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
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3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
Identical,indistinguishable
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3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
Bosons
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3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
3411
Bosons
classical
40%30%20%10%
bosons
33%44%11%11%
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Counting3 particles, total energy = 3
3
012
0123
3
012
1110
fermions
33%33%33%0%
bosons
33%44%11%11%
classical
40%30%20%10%
Fermions
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3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
10 % probabilityfor triple occupancy
30 % probabilityfor double occupancy
Classical
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3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
33 % probabilityfor triple occupancy
33 % probabilityfor double occupancy
Bosons like to get together!
Bosons
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Three different statistics
=)(εn
11
)( +− TkBe με
11
)( −− TkBe με
TkBe )(1με−
classical particlesMaxwell-Boltzmannstatistics
bosonsBose-Einsteinstatistics
fermionsFermi-Diracstatistics
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Thermodynamics of 3-dimensional Ideal Bose Gas.3/ 2
0 1 ,c
TN NTε =
⎛ ⎞⎛ ⎞⎜ ⎟= − ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
2 2 2
2 2 20
12 2
exp 12
k kk
B
k dk kE n Vm k
mk T
επ
∞
= =⎛ ⎞
−⎜ ⎟⎝ ⎠
∑ ∫
5 ,2V
V
CTE
TE∂⎛ ⎞= =⎜ ⎟∂⎝ ⎠
0
53
TVCS dT
TE
T= =∫
F TE S= −
3/ 2 5/ 2
3
( )0.0851 B
T
m k TFPV∂⎛ ⎞= − =⎜ ⎟∂⎝ ⎠
3/ 2 3/ 2 5/ 2
3
( )0.770 0.1289 ,BB
c
m k TTE Nk T VT⎛ ⎞
= =⎜ ⎟⎝ ⎠
Pressure does not depend on the volume
0 0TP →⎯⎯⎯→
23
E= −
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Summary:
All the thermodynamic quantities are continuous at the transition point.
Bose Gas undergoes a phase transition without any interaction!
0N
TcT
cT T−
Order parameterVC
TcT
-point in He4 λ
3rd - order phase transition
2/32
3.31cB
NT Tmk V
⎛ ⎞< = ⎜ ⎟⎝ ⎠
particles start to collect at lowest energy until at T=0 they are all there
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Weakly-interacting Bose Gas
da
Ideal Bose Gas2
( )2ppm
ε =
Weak interactions: a d
1/ 23/ 2
0
813
N NaN Vπ
⎛ ⎞≈ − ⎜ ⎟⎝ ⎠
Interacting Bose Gas
22
222 0
2 202
30
2
2
2 2 30
,2
44( )2
2,
2
pm m
p n an a p mp pm m p
m
n a
n aa
pm ma
ππε
⎧⎪⎛ ⎞ ⎪= + ≈ ⎨⎜ ⎟
⎝ ⎠ ⎪⎪⎩
0
NnV
=
Particles of a non-ideal Bose gas do notall have zero momentum,even in the ground state.
24( )U q amπ
≈
a – scattering length,d – average distance
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Laser lightOrdinary light
diffraction limited (directional)coherentone big wavesingle mode (monochromatic)
divergentincoherentmany small wavesmany modes
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Bose-Einsteincondensate
Ordinary gas
diffraction limited (directional)coherentone big wavesingle mode (monochromatic)
divergentincoherentmany small wavesmany modes
atoms move around randomly atoms in a coherent state
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Possible candidates for the experimental observation of the Bose-Einstein Condensation
2/32
3.312c
BmNT
k V⎛ ⎞= ⎜ ⎟⎝ ⎠
Atomic Hydrogen
Helium Only 8% of particles in the condensate
Strongly interacting Bose system!
Excitons in semiconductorselectron
holeStrong many-body effects
2H
Spin polarized hydrogen
molecularcrystal
U
R
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BEC @ JILA, June ‘95(Rubidium)
BEC @ MIT, Sept. ‘95 (Sodium)
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Rubidium, JILA Group,June 1995
Sodium, MIT Group,September 1995
610∼
310∼
particles in BEC
particles in BEC
1cT Kμ∼
200 270m mμ μ×field of view
JILA –Joint Institute for Laboratory Astrophysics
time – 1/20 s
Dis
tribu
tion
func
tion
Dis
tribu
tion
func
tion
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∝ -1/2dB T=h/p Tλ
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Condensed matter physicsMany-body physicsStatistical physics• Superfluidity• Quantum gases• Mesoscopic physics
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
BEC=Tool for knowledge BEC=Tool for applications
Metrology• Atomic clocks• Matter wave sensors