1. 7. Ideal Bose Systems Thermodynamic Behavior of an Ideal Bose Gas
Bose-Einstein Condensation in Ultracold Atomic Gases
Thermodynamics of the Blackbody Radiation
The Field of Sound Waves
Inertial Density of the Sound Field
Elementary Excitations in Liquid Helium II

2. 7.1. Thermodynamic Behavior of an Ideal Bose Gas
From § 6.1-2, Bose gas :
Grand partition function
Grand potential
 =   BE condensation
DOS

3. Correction for a(0) = 0 :
( to handle )
( See App.F for rigorous justification )
E.g. :
# of particles in ground state :
is negligible
for N0 , V   

4. Bose-Einstein functions

5. = # of particles in
ground state 

6. U, z ( n 3 ) << 1  for z < 1. 
Calculated using Mathematica

7. Virial Coefficients ( z << 1 )
al = Virial Coefficients
= volume per
particle
Calculated using Mathematica

8. CV ( z << 1 )   CV has max. Known :
Calculated using Mathematica 
 CV has max.
Known :

9. z ≤ 1 = density of excited particles   
= # of particles in the ground state N0 1 10
100 z
0.5
0.91
0.99  
BEC ( Bose-Einstein Condensation )

10. Bose-Einstein Condensation ( BEC )
Superconductor : Condensatiion in momentum space
Superfluid : Condensatiion in coordinate space
 Condition for BEC is or
with 
Condensate = mixture of 2 phases :
Normal phase (excited particles)
Condensed phase (ground state p’cles) 
for T <<TC

11. For is obtained by solving For is obtained by solving 
Calculated using Mathematica
For
is obtained by solving
For
is obtained by solving 

12. P ( T ) for all z as V    for T < TC   T < TC  
 ½ PMB (TC)

13. ( Determines z for given n & T. )
for all z
for T  TC
For T > TC , N0 ~ O(1) 
( Determines z for given n & T. )
Calculated using Mathematica
Mixture
( z = 1 ) 
Virial expansions for T >> TC
Inaccessible
( z > 1 )
Bose gas
Classical
Transition line
( P  T 5/2  T )
normal phase ( z < 1)

14. CV  
For T < TC
For T = TC

15. For T < TC
For T > TC    

16. with =
CV / T discontinuous at TC :
Prob.7.6
classical value  

17.  Transition London : He I – He II transition is a BE condensation.
Calculated using Mathematica
m = 6.65  1024 g.
V = 27.6 cm3 / mole
v = V / NA = 4.58  1023
TC = 3.13 K
Exp: TC = 2.19 K
He4
He II
He I

18. Isotherms For isotherms, N, T = const. &
z is a function of v = V / N determined by
Setting 
& z is determined by
for
&

19. Transition line : P( v = vC ) , i.e.,
For v < vC
indep of v
Transition line : P( v = vC ) , i.e.,
T > T
Calculated using Mathematica

20. Adiabats
Fundamental thermodynamic equation :
see Reichl §2.E 

21. Since z = 1 for T > TC , z = const  T for an adiabatic process.
 const z  const n 3
Hence, for an adiabatic process
i.e.  
Same as the ideal classical gas.

22.  Prob 7.4-5  5/3 for T >> TC > 5/3 otherwise   for T = TC
Mixed phase region (T < TC ) : 
( No contribution from N0 )

23. 7.2. Bose-Einstein Condensation in Ultracold Atomic Gases
Magneto-optical traps (MOTs) to cool 104 neutral atoms / molecules at T ~ nK :
Step 1 : T ~  K
3 orthogonal pairs of opposing laser beams with
 Stationary atoms not affected.
Moving atoms Doppler shifted to absorb photon & recoil.
Re-emit photons are isotropic.
 Atoms slowed.
Recoil limit :  

24. Step 2 : T ~ 100 n K
Laser off.
Anisotropic, harmonic potential at trap center created by B(r) .
m = magnetic moment of atom
Evaporative cooling :
 adjusted to resonance to remove highest energy atoms. 
Degeneracy of the level is
Prob 3.26

25. DOS a ( )   

26. Grand Potential ( F =   )
Grand partition function 

27. N  
V = const for a trap
Onset of BEC : z = 1, T = TC, N = Ne = # of trapped atoms.  
For a given T , z is given by

28. T > TC :
Obs. ~ 170 nK
T < TC :
is finite in the TD limit (N ,V   ) .
Occupancy of 1st excited state :
 0 in the TD limit.

29. 7.2.A. Detection of the BEC
Harmonic oscillator : Linear size of ground state along x is
Linear size of thermal distribution of excited atoms is ( equipartition theorem ) : 
For  = 2 ( 100 Hz), T = 100 nK,
Time of flight measurement of momentum distribution f ( p ) :
B turned off  atomic cloud expands for 100 ms according to f ( p ).
( v ~ 1 mm/s  x ~ 100 m. )
Cloud illuminated with laser at resonant  shadow on CCD.
( size & shape of shadow n( r , t ) gives f ( p ) at t = 0 )
3. For long times, n0 ( r , t ) is anisotropic, while ne ( r , t ) is isotropic.

30. n0 For a 1-D harmonic oscillator in its ground state
In the plane wave basis ( p-representation ) : 
Mathematica
At t = 0, B is turned off so that for t > 0, H = p2 / 2m :

31. n anisotropyic for large t ( BEC signature )
Mathematica
t = 0
t > t
t~10ms
n anisotropyic for large t ( BEC signature )

32. nexcited Semi-classical treatment :
ensemble average done in phase space :
BE statistics is used for f :
with
For t > 0

33. n loses anisotropy for large t .
t~10ms
Mathematica

34. 87Rb
Anisotropy is BEC signature.

35. 7.2.B. Thermodynamic Properties of the BEC
Alternatively :
= same result 
for T > TC
Setting z = 1 :
for T < TC
( U = 0 for condensate )

36. Calculated using Mathematica

37. V is const in trap   
T > TC :
T < TC :

38. Calculated using Mathematica