In the quantum level, there are profound differences between fermions (follows Fermi-Dirac statistic) and bosons (follows Bose-Einstein statistic). As a gas of bosonic atoms is cooled very close to absolute zero temperature, their characteristic will change dramatically. More accurately when its temperature below a critical temperature Tc, a large fraction of the atoms condenses in the lowest quantum states . This dramatic phenomenon is known as Bose-Einstein condensation

1. The institute of science
AKSHAY APPA KHORATE
DHIRAJKUMAR HIRALAL HIRALAL

2. Condensation (BEC).
•In the quantum level, there are profound differences
between fermions (follows Fermi-Dirac statistic) and bosons
(follows Bose-Einstein statistic).
•As a gas of bosonic atoms is cooled very close to absolute
zero temperature, their characteristic will change
dramatically.
•More accurately when its temperature below a critical
temperature Tc, a large fraction of the atoms condenses in
the lowest quantum states .
•This dramatic phenomenon is known as Bose-Einstein

3. theories, except at very low temperatures.
• In 1924 an Indian physicist named Bose derived the Planck law
for black-body radiation by treating the photons as a gas of
identical particles.
•Einstein generalized Bose's theory to an ideal gas of identical
atoms or molecules for which the number of particles is
conserved.
•The equations, which were derived by Einstein didn't predict
the behavior of the atoms to be any different from previous

4. •Einstein found that when the temperature is high, they
behave like ordinary gases.
•However, at very low temperatures Einstein's theory
predicted that a significant proportion of the atom in the
gas would collapse into their lowest energy level.
•This is called Bose-Einstein condensation.
•The BEC is essentially a new state of matter where it
is no longer possible to distinguish between the atoms.

5. 1. Ideal Bose gas
The Pauli principle does not apply in this case, and the low-
temperature properties of such a gas are very different
from those of a fermion gas.
The properties of BE gas follow from Bose-Einstein
distribution.
Here T represents the temperature, kb Boltzmann constant
and  the chemical potential.
kBT
k
k
k
n 
1
,n  N ,   1
e (  k )
1

6. In the Bose-Einstein distribution, the number of particles
in the energy range dE is given by n(E)dE, where
g(E)
z1
eE/kBT
n(E) 
1
z is the fugacity, defined by
z e/ kBT
where μ is the chemical potential of the gas, and the density
of states g(E) (which gives the number of states between E
and E+dE) is given (in three dimensions) for volume V by
2m 3/ 2
V
E
g(E) 
42
3

7. The critical (or transition) temperature Tc is defined as the
highest temperature at which there exists macroscopic
occupation of the ground state.
The number of particles in excited states can be calculated
by integrating n(E)d(E):
B
EdE

2m 3/ 2
V
2 3 1 E / k T

Ne
 n(E)dE  
0
4  z e 1
Ne is maximal when z=1 (and thus μ=0), and for a condensate
to exist we require the number of particles in the excited
state to be smaller than the total number of particles N.

8. B
0 0
2m 3/2
V 2mk T3/ 2
V
EdE

3/2 xdx
z1
eE/kBT
 3   3
Ne
   
k T B
   N
 2   2
1    
  ex
42
3
1 42
3
42
3
Therefore
 3   3   2.314
 2   2
   
where
 42
N 
2/3
 Tc
T   
2mk 2.315V
B  
2
Below this temperature most of the atoms will be part of the
BEC.
For example, sodium has a critical temperature of about 2μK.

9. In fact, the condensate fraction, i.e. how many of the
particles are in the BEC, is represented mathematically as,
2
3
0

1 
 TC 
 T 
N
N
where N0 is the number of atoms in the groundstate.
The number of excited particles at temperatures below the
critical temperature can be rewritten as
 T 
3/ 2
Ne
 N 
 Tc
The number of particles at the ground state (and therefore in
the condensate) N0 is given by
e
0
T
  T 
3/ 2

 
N  N  N  N 1 
 
 c 
 

10. T h e system undergoes a phase transition and forms
a Bose-Einstein condensate, where a macroscopic
number of particles occupy the lowest-energy quantum
state.
3
0
2 3
T 2
  
N(T )  N 
2

2

V  3  mkB 
B E C is a phase-transition solely caused by quantum
statistics, in contrast to other phase-transitions (like
melting or crystallization) which depend on the inter-
particle interactions.

12. mk T
B
dB
22
 
dB
 = de Broglie wavelength
m = mass
T = temperature
 = Planck’s constant
Bose-Einstein condensation is based on the wave nature of
particles.
De Broglie proposed that all matter is composed of waves.
Their wavelengths are given by

13. BEC also can be explained as follows, as the atoms are cooled
to these very low temperatures their de Broglie wavelengths
get very large compared to the atomic separation.
Hence, the atoms can no longer be thought of as particles
but rather must be treated as waves.
At everyday temperatures, the de Broglie wavelength is so
small, that we do not see any wave properties of matter, and
the particle description of the atom works just fine.

14. At high temperature, dB is small, and it is very improbable to find two
particles within this distance.
In a simplified quantum description, the atoms can be regarded as
wavepackets with an extension x, approximately given by
Heisenberg’s uncertainty relation x= h/p, where p denotes the
width of the thermal momentum distribution.

15. When the gas is cooled down the de Broglie wavelength
increases.
At the BEC transition temperature, dB becomes comparable
to the distance between atoms, the wavelengths of
neighboring atoms are beginning to overlap and the Bose
condensates forms which is characterized by a
macroscopic population of the ground state of the system.
As the temperature approaches absolute zero, the
thermal cloud disappears leaving a pure Bose condensate.

16. The green line is a phase
boundary. The exact
location of that green
line can move around a
little, but it will be
present for just about
any substance.
Underneath the green line there is a huge area that we
cannot get to in conditions of thermal equilibrium.
It is called the forbidden region.

17. 18

18. Not all particles can have BEC. This is related to the spin of
the particles.
Single protons, neutrons and electrons have a spin of ½.
They cannot appear in the same quantum state. BEC cannot
take place.
Some atoms contain an even number of fermions. They have a
total spin of whole number. They are called bosons.
Example: A 23Na atom has 11 protons, 12 neutrons and 11
electrons.

19. 45

20. • When all the atoms stay in the condensate, all the atoms are
absolutely identical. There is no possible measurement that
can tell them apart.
• Before condensation, the atoms look like fuzzy balls.
• After condensation, the atoms lie exactly on top of
each other (a superatom).

21. There is a drop of condensate at the center.
The condensate is surrounded by uncondensed gas atoms.

22. 59

23. 60

24. 64
When two Bose-Einstein condensates spread out, the interference pattern
reveals their wave nature.

26. 74

28. This is evidence for
condensation of pairs of 6Li
atoms on the BCS side of the
Feshbach resonance.
The condensate fractions
were extracted from images
like these, using a Gaussian
fit function for the ‘‘thermal’’
part and a

29. 77

30. •This is a completely new area. Applications are too early to
predict.
•The atom laser can be used in:
1. atom optics (studying the optical properties of atoms)
2. atom lithography (fabricating extremely small circuits)
3. precision atomic clocks
4. other measurements of fundamental standards hologram
5. communications and computation.
•Fundamental understanding of quantum mechanics. Model
of black holes.

31. Homepage of the Nobel e-Museum (http://www.nobel.se/).
BEC Homepage at the University of Colorado (http://www.colorado.edu
/physics/2000/bec/).
Ketterle Group Homepage (http://www.cua.mit/ketterle_group/).
The Coolest Gas in the Universe (Scientific American, December 2000, 92-99).
Atom Lasers (Physics World, August 1999, 31-35).
http://cua.mit.edu/ketterle_group/Animation_folder/TOFsplit.htm
http://www.colorado.edu/physics/2000/bec/what_it_looks_like.html.
http://www.colorado.edu/physics/2000/bec/lascool4.html.
http://www.colorado.edu/physics/2000/bec/mag_trap.html
Pierre Meystre Atom Optics.