Transcript Document

Bose-Einstein Condensation
Student: Miaoyin Wang
Instructor: Elbio Dagotto
Class: Solid State II, 2010, Spring Semester
Institution: Department of Physics,
University of Tennessee, Knoxville
0. Structure of Presentaton
1.
2.
3.
4.
Bosons and Bose-Einstein Distribution
Bose-Einstein Condensation
Experiment Realization of BEC
Summary
1. Bosons & Bose-Einstein Distribution
Bosons!
Q: How to distinguish one object from another?
- Think of daily life object
-Think of microscopic particles
1. Bosons & Bose-Einstein Distribution
Bosons vs Fermions
Bosons – integer spin / no Pauli exclusion
Fermions – half integer spin / Pauli exclusion
Fermions - 1 way
Classical?
Bosons – 10 ways
1. Bosons & Bose-Einstein Distribution
Bose-Einstein Distribution
For N identical bosons with M available quantum states,
there are
( N  M  1)!
W
N !( M  1)!
ways that the particle can be distributed.
For quantum states with different energies:
( N S  M S  1)!
W  WS  
N S !( M S  1)!
S
S
1. Bosons & Bose-Einstein Distribution
Bose-Einstein Distribution
Consider an ideal gas model for bosons,
we have
2
k2
k 
(2 )3
m3/2 1/2
g ( ) 

2 3
2 h
The entropy of the gas is
S=kB ln W
Thus we can apply Lagrange multipliers
S
U
N
 kB 
 kB 
0
N S
N S
N S
1. Bosons & Bose-Einstein Distribution
Bose-Einstein Distribution
Finally, after some calculation, we get
NS 
1
e
 ( S   )
1
MS
for each energy shell and thus
f BE ( ) 
1
e
 (   )
1
2. Bose-Einstein Condensation (BEC)
Observe the equation:
f BE ( ) 
 
 min  0
1
e
 (   )
 0
1
2. Bose-Einstein Condensation (BEC)
 0
NS 
1
e
 ( S   )
1
MS
Since f(E) is a distribution, it also fulfills
MS
N 1
1
n    N S    ( S   )
V V S
V S e
1


Suppose there is a temperature that
(we are not sure about it!)
Have a try   0
n
2
3
(2m)
3/2


0
 d
1/2
e / kBTc  1
 0
 n 
Tc  

 2.612 
2/3
2
mkB
2
2. Bose-Einstein Condensation (BEC)
What will happen below Tc?
n
2
3
(2m)3/2 

0
 1/2 d 
e(   )/ kBTc  1

Where do the particles go?
n
2
3
(2m)3/2 

0
 1/2 d 
e / kBTc  1

W
( N  M  1)!
N !( M  1)!
2. Bose-Einstein Condensation (BEC)
Where do the particles go?

2
3/2
n  3 (2m)  f ( )  1/2 d 
0
h

2
3/2
n  n0  3 (2m)  f ( )  1/2 d 
0
h
 T 
n0 (T )  n 1   
  Tc 

3/2




f ( ) 
1
e
 (   )
1
 A  (0)
W
( N  M  1)!
N !( M  1)!
3. Experimental BEC
How Tc changes upon n?
 n 
Tc  

2.612


2/3
2 2
mkB
To achieve BEC, one can either decrease the temperature or
increase the particle density.
A temperature-density phase diagram will help a lot!
W
( N  M  1)!
N !( M  1)!
3. Experimental BEC
Difficulty to achieve BEC
Make Time BEC << Time thermal equilibrium
W
( N  M  1)!
N !( M  1)!
3. Experimental BEC
So: have to do it in a hurry
- Pulsed laser beam as detector
- Magneto-Optical Trap (MOT)
Also: make thermal equilibrium time scale larger
- Choice of atoms – Rubidium 87
Still: very very cold!
- MOT cooling
- Cool by expand
W
( N  M  1)!
N !( M  1)!
3. Experimental BEC
Procedure
Normal method
Laser
MOT
Expand
W
( N  M  1)!
N !( M  1)!
3. Experimental BEC
Result: velocity distribution data
W
( N  M  1)!
N !( M  1)!
3. Experimental BEC
Anisotropy of the data
Due to
Heisenberg Uncertainty Principle (h~x*p)
+
Anisotropy of the space distribution
W
( N  M  1)!
N !( M  1)!
3. Experimental BEC
Other type of BEC in Experiment
• Superfluid
~Helium-4 @ 2.17K
~only about 8% of the atoms accumulate in ground state - not a
“pure” BEC.
• Fermions
~extremely low temperature
~must “pair up” to form compound particles (like molecules or
Cooper pairs) that are bosons.
• Magnons
~ a BEC transmission temperature at room temperature
~achieved by pumping the magnons into the system and form a
high density n
W
( N  M  1)!
N !( M  1)!
4. Summary
1. BEC is predicted early and achieved tens of years later,
inspiring a lot of related technologies.
2. BEC in lab is very fragile. Extremely low temperature
and density is required.
3. BEC can be useful in very basic physics.
4. It can also be used in ultra-sensitive measurements.
(Think of laser)
W
( N  M  1)!
N !( M  1)!
5. References
[1] Superconductivity, Superfluids and Condensates, J.F.Annett, ISBN 7-03-023624-1
[2] Thermodynamics and Statistical Mechanics, Zhicheng Wang, ISBN 7-04-011574-3
[3] Levi, Barbara Goss (2001). "Cornell, Ketterle, and Wieman Share Nobel Prize for
Bose–Einstein Condensates". Search & Discovery. Physics Today online.
http://www.physicstoday.org/pt/vol-54/iss-12/p14.html.
[4] Bose-Einstein Condensation, Wikipedia,
http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate#cite_note-nobel-4
[5] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell (1995).
"Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor". Science
269 (5221): 198–201.
[6 ] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker
Denschlag, and R. Grimm (2003). "Bose–Einstein Condensation of Molecules".
Science 302 (5653): 2101–2103
[7] Demokritov, S.O.; Demidov, VE; Dzyapko, O; Melkov, GA; Serga, AA; Hillebrands, B;
Slavin, AN (2006). "Bose–Einstein condensation of quasi-equilibrium magnons at
room temperature under pumping". Nature 443 (7110): 430–433
Thank you!
Student: Miaoyin Wang
Instructor: Elbio Dagotto
Class: Solid State II, 2010, Spring Semester
Institution: Department of Physics,
University of Tennessee, Knoxville