Post on 18-Mar-2018
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Bose-Einstein CondensationM.N.Kiselev
Institut für Theoretische Physik und Astrophysik
Universität Würzburg
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
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
012
0123
12 40%9 30%6 20%3 10%
3
012
Identical, but classically distinguishable
3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
Identical,indistinguishable
3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
Bosons
3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
3411
Bosons
classical
40%30%20%10%
bosons
33%44%11%11%
3 particles, total energy = 3 (Arbitrary units)
3
012
0123
3
012
10 % probabilityfor triple occupancy
30 % probabilityfor double occupancy
Classical
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
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= −
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
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
Laser lightOrdinary light
diffraction limited (directional)coherentone big wavesingle mode (monochromatic)
divergentincoherentmany small wavesmany modes
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
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
BEC @ JILA, June ‘95(Rubidium)
BEC @ MIT, Sept. ‘95 (Sodium)
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
∝ -1/2dB T=h/p Tλ
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