Bose Einstein Condensation

Condensed Matter II –Spring 2007Davi Ortega

In Diluted Gas

Summary

• From counting to a new state of matter– Indistinguishability of particles– Counting indistinguishable particles– Einstein’s conclusion = New State– Some properties

• BEC in Dilute Gases and Liquid Helium• Experiments to achieve the BEC.• News

Statistics of Idea Gas• Two Particles Gas: A and B.

Classical Bosons:Can occupy the same level

Indistinguishable ParticlesFermions:

Cannot occupy the same level

Distinguishable ParticlesAverage number of occupationfor each energy state

(Boson Case)

1)](exp[1

n

Some Theory

Particle in a 3D box:2

2222

22

)(2

L

CmL zyx

Calculating explicitly the total number:

12/3

)(2/3

22 l

lB

leTmkVN

More Theory

1

2/32/3

22 l

cB lTmkVN

But what happens now if I let at this temperature the density n V of the substance increase(e.g., by isothermal compression) to even higher values?

I claim that in this case a number of molecules which always grows with the total density makes a transition to the 1. quantum state (state without kinetic energy)… … The claim thus asserts that something similar happens as when isothermally compressing a vapor beyond the volume of saturation. A separation occurs; a part “condenses”, the rest remains a “saturated ideal gas”. (Einstein, 1925).

Two more useful results

331

2/3

)(...612.2

)( cc

l

TT

l

VN

The order of the de Broglie wavelenght is the same as the volume ocupied by the

ensemble

2/3

0 1cTTNN

Number of the particles that falls to the “condensate” state

An Last statement

This whole theory assume non interacting particles: Schrödinger Equation

Interacting Particles: Gross-Pitaeviski Eq.

)()()()(2

2

int2

2

RERRNVRVM NtrapR

Achievements

• 1938 – Pyotr Kapitsa John Allen and Don Misener: Helium 4

• 1995 – Eric Cornell and Carl Wieman: Rb87: Pure BEC.

• 1995 – Wolfgang Ketterle Na23. (4 months later)

Liquid He vs. Diluted Gas

• Strong Interactions in He qualitatively changes the nature of transition.

• Diluted Gas– Pure Condensate state

• Liquid Helium– Zero Viscosity– And other strange properties.

How to make a BEC ?

• Heat the atoms: 600 K (gas)• Confine them in a beam• Decelerate the beam• Trap the atoms (Magneto Optical Trap)• Magnetic Trapping (avoid atoms recoil by

spontaneous emission and random absorption)

• Evaporative Cooling

Famous Picture

Any other way ?!

YES

Michael ChapmanAll Optical BEC

Current Research

• BEC achieved in several atoms.• Strange Properties:

– Superfluidity– Slowing of light

• Wave-like phenomena in atoms.– Interference between two condensates.

• Atom Laser– Coherent Matter Waves