Images of
Bose-Einstein condensates
Jacek Dziarmaga
& Krzysztof Sacha
PRA 67, 033608 (2003)
cond-mat/0503328
Kraków, Poland
Condensate interference
Bogoliubov theory
Diagonal t-dependent vacuum
Measurements on the vacuum
Javanainen & Yoo, PRL 76, 161 (1996)
Condensate interference
Fock state
N
2
,
N
2
in e
 i x
and
e
 i x
 ( x)   ( x)  ( x)  1

Single density
measurement
Condensate interference
In experiment
P( x1 , x2 ,...., xN ) 






N N
 ( x1 ) ( x2 ).... ( xN )( xN )....( x2 )( x1 )
2 , 2
N
2
, N2
In computer experiment
P( x1 ) 
N
2
,
N
2


 ( x1 ) ( x1 )
N
2
, N2

annihilation of atoms
P( xk 1 | xk ..x1 ) 






N N
N N
,

(
x
)..

(
x
)

(
x
)

(
x
)

(
x
)..

(
x
)
1
k
k 1
k 1
k
1 2 , 2
2 2




Condensate interference
2
N
2
,
N
2

 d
 i 2
ix
e  e
i 2
1
2
N: e
1
2
N : cos( x  2 )
e ix

0
2
 d
0


 ( xk ).... ( x1 )
2
N
2
,
N
2

 d
 ( , 0 )

2
N : cos( x  )
0
  0 ( x1 ,...., xk )
1
 
k
Phase ``collapse’’ by
density measurement
Quantum Bogoliubov theory
Hamiltonian

H   dx
















1
1





V
(
x
)







x
2 x
2 g  
Small fluctuations

 ( x) 


N 0 ( x)  ( x)
 0     V ( x)  g  0  H GP 0
1
2
2
x
*
0 0


 
 
H 2   dx  , L    
 

with

 H GP  g0*0
L  
* *

g

0 0

Expansion
Fig.Jav
g0 0


 H GP  g0*0 
Bogoliubov transformation

 ( x) 


m 1

 *
bm um ( x)  bm vm ( x)
 um 
m    L
 vm 


bm  um 
 um 
 
 vm 

 vm 
Bogoliubov Hamiltonian

 

H 2    m bm bm
m 1
particles <-> quasiparticles
Bogoliubov vacuum

bm 0b  0
Time-dependent Bogoliubov theory
Condensate in the ground state
 0b
V ( x, t )
Condensate in an excited state


 ( x )  N 0 ( x )   ( x )

 0b ( t )

i  t  0       g   V ( x , t )  0
1
2
2
x
*
0 0
Time-dependent Bogoliubov theory

 


m 1

 *
bm um (t )  bm vm (t )
0b 
 0b
t
Excited state =
t-dependent vacuum
Ansatz
 um 
 um 
i t    L(t )   Solution
 vm 
 vm 



bm (t )  um (t )   vm (t ) 

bm (t ) 0b ( t )  0
Diagonal dynamical vacuum
We claim
0b ( t )
N
2
  
 
  a0 a0    a a  0
 1



0 (t )  condensate wave function
 (t )  [0,1)
 (t )  orthonorma l basis
D & Sacha, PRA 67, 033608 (2003)
cond-mat/0503328
Proof (constructive)


0b ( t )  ( x) ( y ) 0b ( t )






1
dN  (t , x)  (t , y )
*
0b ( t )  ( x) ( y ) 0b ( t ) 



e
2 i 
1
 (t ) 
dN (1  dN )  (t , x)  (t , y )
dN
 [0,1)
1  dN
 (t , x)   (t , x) ei 
um (t ) vm (t )   (t )  (t )
q - representation
  
 
 a0 a0    a a 




dq
exp




 
 
1 
2 
N
2
0

q 

2

N : 0 
1
N
q 


Real coordinates q
q - representation

 dq exp   
 
1
2 
For dN 
q q 
2
1 2
e
P(q)  e


q 

 12
N : 0 
2

( q  q )
  q
1

q 


 1


1
N
2
  (q  q)
2

 e
 12


q2
dN
Probability for q
Density measurement
P(q) 



exp 
q2
1
2 dN
 ( x | q )  N  0( x ) 

1
N

q  ( x )


2
Dark soliton
i t0      x 0  g 0 0  V (t , x)0
1
2
2
x 0
1
2
2
2
Fig:imprinting
Phase imprinting
Organizers
Burger et al., PRL 83, 5198 (1999)
Phase imprinting in Bogoliubov theory
Condesate
Density
after 2ms
In focus
Total
density
after 2ms
Mode 1
Density measurement

0b  a a   a a
 
0 0
N  1.5 105
 
1 1 1

N
2
0 Truncated vacuum
dN1  273
Solid line : N 0 
with q1  14.48

N 1
q1
2
Conclusion
0b
  
 
  a0 a0    a a 



N
2
0
Condensate in thermal state
V ( x, t )
Condensate in ``excited thermal state’’

 12 bL* bL  12 bR* bR bL* e   bR
2
2
   d bL d bR e
N : bL N : bR
  

Tr   ( x) ( y )
      ( x | q)  N 0 (t , x)   q  (t , x)
Tr   ( x) ( y) 
 1
 (t , x) are NOT orthogonal