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BEC is the Holy-Grail
of atomic physics!
Why has it taken so long to make BEC?
3dB  2.612
Liquid helium
Derivation of Gross-Pitaevskii equation
The grand-canonical Hamiltonian for an interacting many-body
system with total particle number N is given by
Kˆ  Hˆ   Nˆ
2 2

 ˆ

†
ˆ
  dx  x   
 Vext  x       x 
 2m

1
ˆ † y 
ˆ †  x V  x, y 
ˆ x 
ˆ y 
  dxdy
2
where
ˆ  x  : quantum field for the particle

 : chemical potential
and
ˆ † x 
ˆ x  N
d
x


Bogoliubov decomposition for bosons
ˆ  x     x   ˆ  x 

ˆ  x  : macroscopic wave function
 x  
ˆ  x  : field operator for noncondensed particle
Mean-field approximations
ˆ †  y ˆ †  x ˆ  x ˆ  y 
ˆ †  y ˆ †  x  ˆ  x ˆ  y   ˆ †  y ˆ  x  ˆ †  x ˆ  y 
 ˆ †  y ˆ  y  ˆ †  x ˆ  x   ˆ †  x ˆ  x  ˆ †  y ˆ  y 
 ˆ †  x ˆ  y  ˆ †  y ˆ  x   ˆ  x ˆ  y  ˆ †  y ˆ †  x 
ˆ †  y ˆ †  x    x ˆ  y 
  x   ˆ †  y ˆ †  x  ˆ  y   ˆ †  y ˆ  y  ˆ †  x   ˆ †  x ˆ  y  ˆ †  y  
†
ˆ
ˆ


and other terms cubic in and
Consequently, we have the simplified grand-canonical Hamiltonian
Kˆ eff  K0  Kˆ1  Kˆ 1†  Kˆ 2
where the energy functional for the condensate
2 2



K 0   dx*  x   
 Vext  x       x 
 2m

1
  dxdy*  y  *  x  V  x, y    x    y 
2
is independent of ˆ , and K̂1 and K̂2 are linear and quadratic
in ˆ respectively.
Now consider the short-range -interaction
4 2 a
V  x, y  
 x  y
m
where a is the s-wave scattering length.
After minimizing K0 with respect to the ground state wave function,
one obtains the time-independent Gross-Pitaevskii equation (GPE)
2
 2 2
4 2 a
 Vext  x  
  x     x     x 

m
 2m

The time-dependent GPE can be obtained straightforwardly
2
 2 2

4 2 a
i
  x, t   
 Vext  x  
  x, t     x, t 
t
m
 2m

Particle-in-a-box picture of the scattering length
 
Eg 
 
2m  L 
2
Eg 
  


2m  L  A 
2
Consider a pair of atoms with
reduced mass m  m / 2, whose
relative motion is confined to a
box of lengh L.
2
2
Eg 
A
mL3
For N pairs
  
Eg 


2m  L  A 
2
2
Eg 
A
mL3
total energy difference
AN
A

 n
3
mL
m