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